Finite Group Databasebeta

← C2×#22558337 #30770821

Subgroups — C2×#22558337 #30770821

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863380C211yesmaximal, characteristic
2C2×#16863363C211yesmaximal, characteristic
3#22558337C214yesmaximal, direct product factor
4C2×#410×C67C211yesmaximal, characteristic
5C22×#11761612—21no—
6C22×#11761612C411yescharacteristic, complemented
7C23×C67⋊C8C411yescharacteristic, complemented
8#16863380—21no—
9#16863380—21no—
10#16863380C412yescomplemented
11#16863380C2212yes—
12C2×#14523962C412yescomplemented
13C2×#14523960—21no—
14#16863363—21no—
15#16863363—21no—
16#16863363C412yescomplemented
17#16863363C2212yes—
18C2×C22⋊(C67⋊C8)—21no—
19C2×C22⋊(C67⋊C8)—21no—
20C22×#41×C67C2211yescharacteristic
21C2×#117×C67—21no—
22#410×C67—21no—
23#410×C67—21no—
24#410×C67C412yes—
25#410×C67C2212yes—
26C2×#99×C67—21no—
27C2×#99×C67C412yes—
28C2×#11761612—21no—
29C2×#11761612—22no—
30C2×#11761612—22no—
31C2×#11761612—22no—
32C2×#11761612—41no—
33C2×#11761612—41no—
34C2×#11761612—41no—
35C2×#11761612—41no—
36C2×#11761612—41no—
37C2×#11761612C2×C412yescomplemented
38C22×C67⋊C8—21no—
39C22×C67⋊C8—21no—
40C22×C67⋊C8—21no—
41C22×C67⋊C8—41no—
42C22×C67⋊C8—41no—
43C22×C67⋊C8—41no—
44C22×C67⋊C8C2×C412yescomplemented
45#14523962—22no—
46#14523962C2×C414yescomplemented
47#14523960—22no—
48#14523960—22no—
49C22⋊(C67⋊C8)—22no—
50C22⋊(C67⋊C8)—22no—
51C22⋊(C67⋊C8)—22no—
52C22⋊(C67⋊C8)—22no—
53C2×#41×C67—41no—
54C2×#41×C67—41no—
55C2×#41×C67—41no—
56C2×#41×C67—41no—
57C2×#41×C67C2×C411yescharacteristic
58C2×#41×C67C2×C411yescharacteristic
59C2×#41×C67C2×C411yescharacteristic
60C2×#41×C67D411yescharacteristic
61C2×#41×C67D411yescharacteristic
62C2×#41×C67D411yescharacteristic
63C2×#41×C67Q811yescharacteristic
64C2×#41×C67C2311yescharacteristic
65C22×Q8×C67C2×C411yescharacteristic
66C22×D4×C67C2×C411yescharacteristic
67C22×D4×C67D411yescharacteristic
68C22×D4×C67Q811yescharacteristic
69C23×C4×C67C2×C411yescharacteristic
70C23×C4×C67D411yescharacteristic
71C23×C4×C67D411yescharacteristic
72#117×C67—22no—
73#117×C67—22no—
74#117×C67—41no—
75#117×C67—41no—
76C2×C4⋊C4×C67—41no—
77C2×C22⋊C4×C67—22no—
78C2×C42×C67—41no—
79#99×C67—22no—
80#99×C67—22no—
81#99×C67—22no—
82#99×C67C2×C414yes—
83#11761612—24no—
84#11761612—42no—
85#11761612—42no—
86#11761612—42no—
87#11761612—42no—
88#11761612—42no—
89C2×C67⋊C8—22no—
90C2×C67⋊C8—22no—
91C2×C67⋊C8—42no—
92C2×C67⋊C8—42no—
93C2×C67⋊C8—42no—
94C2×C67⋊C8—42no—
95C2×C67⋊C8—41no—
96C2×C67⋊C8—41no—
97C2×C67⋊C8—41no—
98C2×C67⋊C8—41no—
99C2×C67⋊C8—42no—
100C24×C67C22⋊C411yescharacteristic
101C24×C67C4⋊C411yescharacteristic
102C24×C67C4⋊C411yescharacteristic
103#41×C67—41no—
104#41×C67—41no—
105#41×C67—41no—
106#41×C67—41no—
107#41×C67—41no—
108#41×C67—41no—
109#41×C67—41no—
110#41×C67—41no—
111#41×C67—41no—
112#41×C67—41no—
113#41×C67—41no—
114#41×C67—41no—
115#41×C67—41no—
116#41×C67—41no—
117#41×C67—41no—
118#41×C67—41no—
119C2×Q8×C67—41no—
120C2×Q8×C67—41no—
121C2×Q8×C67C4211yescharacteristic
122C2×Q8×C67C22⋊C411yescharacteristic
123C2×Q8×C67C4⋊C411yescharacteristic
124C2×Q8×C67C22×C411yescharacteristic
125C2×D4×C67—41no—
126C2×D4×C67—41no—
127C2×D4×C67—41no—
128C2×D4×C67—41no—
129C2×D4×C67—41no—
130C2×D4×C67—41no—
131C2×D4×C67C4211yescharacteristic
132C2×D4×C67C22⋊C411yescharacteristic
133C2×D4×C67C22⋊C411yescharacteristic
134C2×D4×C67C22⋊C411yescharacteristic
135C2×D4×C67C4⋊C411yescharacteristic
136C2×D4×C67C4⋊C411yescharacteristic
137C2×D4×C67C4⋊C411yescharacteristic
138C2×D4×C67C4⋊C411yescharacteristic
139C2×D4×C67C4⋊C411yescharacteristic
140C2×D4×C67C22×C411yescharacteristic
141C2×D4×C67C2×D411yescharacteristic
142C2×D4×C67C2×Q811yescharacteristic
143C22×C4×C67—21no—
144C22×C4×C67—21no—
145C22×C4×C67—21no—
146C22×C4×C67—21no—
147C22×C4×C67—21no—
148C22×C4×C67—21no—
149C22×C4×C67—21no—
150C22×C4×C67—21no—
151C22×C4×C67—21no—
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—42no—
156C22×C4×C67C4211yescharacteristic
157C22×C4×C67C4211yescharacteristic
158C22×C4×C67C22⋊C411yescharacteristic
159C22×C4×C67C22⋊C411yescharacteristic
160C22×C4×C67C22⋊C411yescharacteristic
161C22×C4×C67C22⋊C411yescharacteristic
162C22×C4×C67C22⋊C411yescharacteristic
163C22×C4×C67C22⋊C411yescharacteristic
164C22×C4×C67C22⋊C411yescharacteristic
165C22×C4×C67C4⋊C411yescharacteristic
166C22×C4×C67C4⋊C411yescharacteristic
167C22×C4×C67C4⋊C411yescharacteristic
168C22×C4×C67C4⋊C411yescharacteristic
169C22×C4×C67C22×C411yescharacteristic
170C22×C4×C67C2×D411yescharacteristic
171C22×C4×C67C2×D411yescharacteristic
172C4⋊C4×C67—42no—
173C4⋊C4×C67—42no—
174C22⋊C4×C67—24no—
175C22⋊C4×C67—42no—
176C42×C67—42no—
177C42×C67—42no—
178C23×C67—21no—
179C23×C67—21no—
180C23×C67—21no—
181C23×C67—21no—
182C23×C67—21no—
183C23×C67—21no—
184C23×C67—42no—
185C23×C67—42no—
186C23×C67—42no—
187C23×C67#9511yescharacteristic
188C23×C67#9511yescharacteristic
189C23×C67#9511yescharacteristic
190C23×C67#9511yescharacteristic
191C23×C67C2×C22⋊C411yescharacteristic
192C23×C67C2×C4⋊C411yescharacteristic
193C23×C67C2×C4⋊C411yescharacteristic
194Q8×C67—41no—
195Q8×C67—41no—
196Q8×C67—41no—
197Q8×C67—41no—
198D4×C67—41no—
199D4×C67—41no—
200D4×C67—41no—
201D4×C67—41no—
202D4×C67—41no—
203D4×C67—41no—
204D4×C67—41no—
205D4×C67—41no—
206D4×C67—41no—
207D4×C67—41no—
208D4×C67—41no—
209D4×C67—41no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—21no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—44no—
241C2×C4×C67—84no—
242C2×C4×C67#9511yescharacteristic
243C2×C4×C67#9511yescharacteristic
244C2×C4×C67#9511yescharacteristic
245C2×C4×C67#9511yescharacteristic
246C2×C4×C67#9911yescharacteristic
247C2×C4×C67#9911yescharacteristic
248C2×C4×C67#9911yescharacteristic
249C2×C4×C67#9911yescharacteristic
250C2×C4×C67C2×C4211yescharacteristic
251C2×C4×C67C2×C22⋊C411yescharacteristic
252C2×C4×C67C2×C22⋊C411yescharacteristic
253C2×C4×C67C2×C4⋊C411yescharacteristic
254C67⋊C8—44no—
255C67⋊C8—42no—
256C67⋊C8—42no—
257C22×C67—22no—
258C22×C67—21no—
259C22×C67—21no—
260C22×C67—21no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—21no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—42no—
271C22×C67—42no—
272C22×C67—42no—
273C22×C67—42no—
274C22×C67—81no—
275C22×C67—81no—
276C22×C67#34211yescharacteristic
277C22×C67#34211yescharacteristic
278C22×C67C2×#9511yescharacteristic
279C4×C67—21no—
280C4×C67—21no—
281C4×C67—21no—
282C4×C67—21no—
283C4×C67—21no—
284C4×C67—21no—
285C4×C67—84no—
286C4×C67#34211yescharacteristic
287C4×C67#34211yescharacteristic
288C4×C67C2×#9911yescharacteristic
289C4×C67C2×#9911yescharacteristic
290C2×#1395—671nomaximal, Sylow
291C2×C67—21no—
292C2×C67—21no—
293C2×C67—41no—
294C2×C67—41no—
295C2×C67—41no—
296C2×C67—41no—
297C2×C67—41no—
298C2×C67—41no—
299C2×C67#139512yescomplemented
300C2×C67C2×#34211yescharacteristic
301C2×#413—671no—
302C2×#410D6711yescharacteristic
303C2×#408—671no—
304#1395—674no—
305C67C2×#139511yescharacteristic, complemented, Sylow
306C22×#41C2×D6711yescharacteristic
307C22×M16—671no—
308C22×M16—1341no—
309C23×C8—671no—
310C2×#117—21no—
311#413—672no—
312#413—672no—
313#413—1341no—
314#413—1341no—
315C2×#101—672no—
316C2×#100—1341no—
317#410—21no—
318#410—21no—
319#410Dic6712yes—
320#410C2×D6712yes—
321C2×#99—21no—
322C2×#99Dic6712yes—
323#408—672no—
324#408—672no—
325#408—1341no—
326#408—1341no—
327C2×C22⋊C8—1341no—
328C2×C22⋊C8—1341no—
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic13411yescharacteristic
334C2×#41C4×D6711yescharacteristic
335C2×#41C4×D6711yescharacteristic
336C2×#41D26811yescharacteristic
337C2×#41C2×Dic6711yescharacteristic
338C2×#41C22⋊D6711yescharacteristic
339C2×#41C22⋊D6711yescharacteristic
340C2×#41C22×D6711yescharacteristic
341C22×Q8C4×D6711yescharacteristic
342C22×D4Dic13411yescharacteristic
343C22×D4C2×Dic6711yescharacteristic
344C22×D4C22⋊D6711yescharacteristic
345C23×C4C4×D6711yescharacteristic
346C23×C4D26811yescharacteristic
347C23×C4C22⋊D6711yescharacteristic
348C2×M16—2681no—
349C2×M16—2681no—
350C2×M16—2681no—
351C2×M16—2681no—
352C2×M16—2681no—
353C2×M16—672no—
354C2×M16—1342no—
355C2×M16—1342no—
356C2×M16—1342no—
357C2×M16—1341no—
358C22×C8—2681no—
359C22×C8—2681no—
360C22×C8—2681no—
361C22×C8—672no—
362C22×C8—1341no—
363C22×C8—1341no—
364C22×C8—1341no—
365#117—22no—
366#117—22no—
367#117—41no—
368#117—41no—
369C2×C4⋊C4—41no—
370C2×C22⋊C4—22no—
371C2×C42—41no—
372#101—674no—
373#101—1342no—
374#100—1342no—
375#100—1342no—
376#99—22no—
377#99—22no—
378#99—22no—
379#99C2×Dic6714yes—
380C22⋊C8—1342no—
381C22⋊C8—1342no—
382C22⋊C8—1342no—
383C22⋊C8—1342no—
384C24C22⋊Dic6711yescharacteristic
385C24Dic67⋊C411yescharacteristic
386C24C4⋊Dic6711yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic6711yescharacteristic
406C2×Q8(C2×D67)⋊C411yescharacteristic
407C2×Q8Dic67⋊C411yescharacteristic
408C2×Q8C2×C4×D6711yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic6711yescharacteristic
416C2×D4(C2×D67)⋊C411yescharacteristic
417C2×D4C22⋊Dic6711yescharacteristic
418C2×D4C22⋊Dic6711yescharacteristic
419C2×D4Dic67⋊C411yescharacteristic
420C2×D4Dic67⋊C411yescharacteristic
421C2×D4Dic67⋊C411yescharacteristic
422C2×D4C4⋊Dic6711yescharacteristic
423C2×D4C4⋊Dic6711yescharacteristic
424C2×D4C22×Dic6711yescharacteristic
425C2×D4C2×C22⋊D6711yescharacteristic
426C2×D4C2×Dic13411yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4C4×Dic6711yescharacteristic
441C22×C4C4×Dic6711yescharacteristic
442C22×C4(C2×D67)⋊C411yescharacteristic
443C22×C4(C2×D67)⋊C411yescharacteristic
444C22×C4(C2×D67)⋊C411yescharacteristic
445C22×C4(C2×D67)⋊C411yescharacteristic
446C22×C4(C2×D67)⋊C411yescharacteristic
447C22×C4(C2×D67)⋊C411yescharacteristic
448C22×C4C22⋊Dic6711yescharacteristic
449C22×C4Dic67⋊C411yescharacteristic
450C22×C4Dic67⋊C411yescharacteristic
451C22×C4Dic67⋊C411yescharacteristic
452C22×C4C4⋊Dic6711yescharacteristic
453C22×C4C2×C4×D6711yescharacteristic
454C22×C4C2×C22⋊D6711yescharacteristic
455C22×C4C2×D26811yescharacteristic
456M16—2682no—
457M16—2682no—
458M16—2682no—
459M16—2682no—
460M16—2682no—
461M16—1344no—
462C2×C8—2682no—
463C2×C8—2682no—
464C2×C8—2682no—
465C2×C8—2682no—
466C2×C8—2681no—
467C2×C8—2681no—
468C2×C8—2681no—
469C2×C8—2681no—
470C2×C8—2682no—
471C2×C8—1342no—
472C2×C8—1342no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C42—42no—
478C42—42no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C67⋊#9511yescharacteristic
489C23C67⋊#9511yescharacteristic
490C23C67⋊#9511yescharacteristic
491C23C67⋊#9511yescharacteristic
492C23C2×C22⋊Dic6711yescharacteristic
493C23C2×Dic67⋊C411yescharacteristic
494C23C2×C4⋊Dic6711yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—84no—
543C2×C4C67⋊#9511yescharacteristic
544C2×C4C67⋊#9511yescharacteristic
545C2×C4C67⋊#9511yescharacteristic
546C2×C4C67⋊#9511yescharacteristic
547C2×C4#1452395711yescharacteristic
548C2×C4#1452395711yescharacteristic
549C2×C4#1452395711yescharacteristic
550C2×C4#1452395711yescharacteristic
551C2×C4C2×C4×Dic6711yescharacteristic
552C2×C4C2×(C2×D67)⋊C411yescharacteristic
553C2×C4C2×(C2×D67)⋊C411yescharacteristic
554C2×C4C2×Dic67⋊C411yescharacteristic
555C8—2684no—
556C8—2682no—
557C8—2682no—
558C22—22no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22#1686316211yescharacteristic
578C22#1686316211yescharacteristic
579C22C2×C67⋊#9511yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—84no—
587C4#1686316211yescharacteristic
588C4#1686316211yescharacteristic
589C4C2×#1452395711yescharacteristic
590C4C2×#1452395711yescharacteristic
591C2—21no—
592C2—21no—
593C2—41no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2#2255833712yesdirect product factor
600C2C2×#1686316211yescharacteristic
subgroupG/H
centerC2×C4#14523957
commutatorC2×C4×C67C2×C42
FrattiniC2×#41C22×D67
FittingC2×#410×C67C2
socleC22×C67#342
prime radicalC22×#41×C67C22
cosocleC2×#41×C67C23
hypercenterC2×#410D67
hypocenterC67C2×#1395
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22558337C2direct (N × H)
6. C22×#11761612C4semidirect
7. C23×C67⋊C8C4semidirect
10. #16863380C4semidirect
12. C2×#14523962C4semidirect
16. #16863363C4semidirect
37. C2×#11761612C2×C4semidirect
44. C22×C67⋊C8C2×C4semidirect
46. #14523962C2×C4semidirect
299. C2×C67#1395semidirect
305. C67C2×#1395semidirect
599. C2#22558337direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168633802
C2×#168633632
#225583372
C2×#410×C672
C2×#139567