Finite Group Databasebeta

← C2×#11730933 #13521816

Subgroups — C2×#11730933 #13521816

Order 1792.

3246 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C22×#6487C211yesmaximal, characteristic
3C22×#6432C211yesmaximal, characteristic, complemented
4C22×C4×Dic14C211yesmaximal, characteristic, complemented
5C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
6C2×#89720C212yesmaximal
7C2×#89723C212yesmaximal
8C2×#89701C211yesmaximal, characteristic, complemented
9C2×#89700C211yesmaximal, characteristic, complemented
10#11730933C2116yesmaximal, direct product factor
11C2×#89429C212yesmaximal, complemented
12C2×#89419C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C23×Dic14C2211yescharacteristic
15C23×Dic14C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C22×C22⋊Dic7C2212yes—
21C22×C22⋊Dic7C2212yes—
22C22×C4⋊Dic7C2212yes—
23C22×C4⋊Dic7C2212yes—
24C22×C4⋊Dic7C2211yescharacteristic
25C22×C4⋊Dic7C2211yescharacteristic
26C2×#6487—28no—
27C2×#6487—28no—
28C2×#6487C2218yes—
29C2×#6487C2218yes—
30C22×Dic7⋊C4C2212yes—
31C22×Dic7⋊C4C2212yes—
32C22×Dic7⋊C4C2212yes—
33C22×Dic7⋊C4C2212yes—
34C22×C4×Dic7C2212yes—
35C2×#6432—28no—
36C2×#6432C2218yescomplemented
37C2×C4×Dic14—28no—
38C2×C4×Dic14C2218yescomplemented
39C4×C7×C22⋊C4C2218yes—
40C2×C7×#95C2211yescharacteristic
41C2×C7×#95C2211yescharacteristic
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45#89723C22116yes—
46#89720C22116yes—
47#89701C2218yescomplemented
48#89700C2218yescomplemented
49#89429C22116yescomplemented
50#89419C22116yescomplemented
51C2×#2429—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C23×C4×C7—24no—
54C23×C4×C7—24no—
55C23×C4×C7—24no—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C23×Dic7C2312yes—
64C23×Dic7C2312yes—
65C23×Dic7C2312yes—
66C23×Dic7C2312yes—
67C22×Dic14—24no—
68C22×Dic14—24no—
69C22×Dic14—24no—
70C22×Dic14—24no—
71C22×Dic14—24no—
72C22×Dic14D418yes—
73C22×Dic14C2314yes—
74C22×Dic14C2314yes—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4Q818yescomplemented
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2314yes—
79C2×C42×C7—24no—
80C2×C42×C7—24no—
81C2×C42×C7C2314yes—
82C2×C42×C7C2314yes—
83C2×C22⋊Dic7—28no—
84C2×C22⋊Dic7—28no—
85C2×C22⋊Dic7C2318yes—
86C2×C22⋊Dic7C2318yes—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7—28no—
89C2×C4⋊Dic7—28no—
90C2×C4⋊Dic7D418yes—
91C2×C4⋊Dic7C2318yes—
92C2×C4⋊Dic7C2318yes—
93C2×C4⋊Dic7C2314yes—
94C2×C4⋊Dic7C2314yes—
95#6487—232no—
96#6487—232no—
97C2×Dic7⋊C4—28no—
98C2×Dic7⋊C4—28no—
99C2×Dic7⋊C4—28no—
100C2×Dic7⋊C4—28no—
101C2×Dic7⋊C4C2318yes—
102C2×Dic7⋊C4C2318yes—
103C2×Dic7⋊C4C2318yes—
104C2×Dic7⋊C4C2318yes—
105C2×C4×Dic7—28no—
106C2×C4×Dic7C2318yes—
107#6432—232no—
108C4×Dic14—232no—
109C7×#95C2314yes—
110C7×#95C2314yes—
111C7⋊#95C2318yes—
112C7⋊#95C2318yes—
113C7⋊#95C2318yes—
114C22×C4⋊Q8—71no—
115C22×C22⋊Q8—71no—
116C22×C22⋊Q8—71no—
117C22×C4×Q8—71no—
118#2429—716no—
119C2×#398—72no—
120C2×#396—72no—
121C2×#391—71no—
122C2×#389—71no—
123C2×#389—72no—
124C2×#388—72no—
125C2×C4×C22⋊C4D711yescharacteristic
126C24×C7—22no—
127C24×C7—22no—
128C24×C7—22no—
129C24×C7—22no—
130C24×C7—22no—
131C24×C7—22no—
132C24×C7C2×Q814yes—
133C24×C7C2412yes—
134C24×C7C2411yescharacteristic
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—28no—
148C22×C4×C7—28no—
149C22×C4×C7—28no—
150C22×C4×C7—416no—
151C22×C4×C7C2×D414yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×Q814yes—
154C22×C4×C7C2×Q814yes—
155C22×C4×C7#4114yes—
156C22×C4×C7#4114yes—
157C22×C4×C7C2412yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2412yes—
161C22×C4×C7C2412yes—
162C22×C4×C7C2412yes—
163C22×Dic7—24no—
164C22×Dic7—24no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—24no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7—24no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7C2×D418yes—
174C22×Dic7C2×D418yes—
175C22×Dic7C2414yes—
176C22×Dic7C2414yes—
177C22×Dic7C2414yes—
178C22×Dic7C2414yes—
179C2×Dic14—232no—
180C2×Dic14—28no—
181C2×Dic14—28no—
182C2×Dic14—28no—
183C2×Dic14—28no—
184C2×Dic14—28no—
185C2×Dic14—416no—
186C2×Dic14C2×D4116yes—
187C7×C22⋊C4—28no—
188C7×C22⋊C4C2×Q8116yescomplemented
189C42×C7—28no—
190C42×C7—28no—
191C22⋊Dic7—216no—
192C22⋊Dic7—216no—
193C4⋊Dic7—28no—
194C4⋊Dic7—216no—
195C4⋊Dic7—216no—
196C4⋊Dic7C2×D4116yes—
197Dic7⋊C4—216no—
198Dic7⋊C4—216no—
199Dic7⋊C4—216no—
200Dic7⋊C4—216no—
201C4×Dic7—216no—
202C23×Q8—71no—
203C23×Q8—71no—
204C24×C4C2×D711yescharacteristic
205C2×C4⋊Q8—78no—
206C2×C4⋊Q8—148no—
207C2×C22⋊Q8—78no—
208C2×C22⋊Q8—78no—
209C2×C22⋊Q8—148no—
210C2×C22⋊Q8—148no—
211C2×C4×Q8—78no—
212C2×C4×Q8—148no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—72no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—72no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—71no—
221C22×C22⋊C4—72no—
222C22×C22⋊C4—72no—
223C22×C22⋊C4C2×D711yescharacteristic
224C22×C22⋊C4C2×D711yescharacteristic
225C22×C42—72no—
226C22×C42C2×D711yescharacteristic
227C22×C42C2×D711yescharacteristic
228#398—716no—
229#396—716no—
230#391—78no—
231#389—78no—
232#389—716no—
233#388—716no—
234C4×C22⋊C4C2×D718yes—
235C2×#95—72no—
236C2×#95—72no—
237C2×#95—72no—
238C2×#95C2×D711yescharacteristic
239C2×#95C2×D711yescharacteristic
240C23×C7—28no—
241C23×C7—28no—
242C23×C7—28no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—22no—
249C23×C7—44no—
250C23×C7—44no—
251C23×C7—44no—
252C23×C7—44no—
253C23×C7C22×D412yes—
254C23×C7C22×D412yes—
255C23×C7C22×Q814yes—
256C23×C7C22×Q812yes—
257C23×C7C2×#4112yes—
258C23×C7C2×#4112yes—
259C23×C72−1+412yes—
260C23×C72−1+412yes—
261C23×C7C2511yescharacteristic
262C2×C4×C7—28no—
263C2×C4×C7—28no—
264C2×C4×C7—28no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—44no—
278C2×C4×C7—44no—
279C2×C4×C7—416no—
280C2×C4×C7#121116yes—
281C2×C4×C7C22×D414yes—
282C2×C4×C7C22×D414yes—
283C2×C4×C7C22×Q814yes—
284C2×C4×C7C22×Q814yes—
285C2×C4×C7C2×#4114yes—
286C2×C4×C7C2×#4114yes—
287C2×Dic7—216no—
288C2×Dic7—216no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—48no—
300C2×Dic7—48no—
301C2×Dic7C22×D418yes—
302C2×Dic7C22×D418yes—
303Dic14—232no—
304Dic14—416no—
305C25C22×D711yescharacteristic
306C22×Q8—78no—
307C22×Q8—74no—
308C22×Q8—74no—
309C22×Q8—144no—
310C22×Q8—144no—
311C22×Q8—144no—
312C22×Q8—144no—
313C22×Q8—144no—
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—24no—
317C23×C4—72no—
318C23×C4—72no—
319C23×C4—72no—
320C23×C4—72no—
321C23×C4C22×D714yes—
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D711yescharacteristic
324C23×C4C22×D711yescharacteristic
325C23×C4C22×D711yescharacteristic
326C23×C4C22×D711yescharacteristic
327C23×C4C22×D711yescharacteristic
328C4⋊Q8—1432no—
329C22⋊Q8—1432no—
330C22⋊Q8—1432no—
331C4×Q8—1432no—
332C2×C4⋊C4—78no—
333C2×C4⋊C4—74no—
334C2×C4⋊C4—78no—
335C2×C4⋊C4—78no—
336C2×C4⋊C4—78no—
337C2×C4⋊C4—74no—
338C2×C4⋊C4—78no—
339C2×C4⋊C4—78no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—148no—
342C2×C4⋊C4—148no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—148no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—148no—
347C2×C4⋊C4—144no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—78no—
350C2×C22⋊C4—78no—
351C2×C22⋊C4—148no—
352C2×C22⋊C4—148no—
353C2×C22⋊C4Dic1418yescomplemented
354C2×C22⋊C4C22×D714yes—
355C2×C22⋊C4C22×D714yes—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—78no—
359C2×C42—148no—
360C2×C42C22×D714yes—
361C2×C42C22×D714yes—
362#95—78no—
363#95—78no—
364#95—78no—
365#95C22×D714yes—
366#95C22×D714yes—
367C22×C7—24no—
368C22×C7—24no—
369C22×C7—24no—
370C22×C7—42no—
371C22×C7—416no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7C2×#12114yes—
376C22×C7#53314yes—
377C22×C7#54114yes—
378C22×C7#54714yes—
379C22×C7#54814yes—
380C22×C7D4×Q814yes—
381C22×C7D4×Q814yes—
382C22×C7C23×D411yescharacteristic
383C22×C7C23×D411yescharacteristic
384C22×C7C23×Q811yescharacteristic
385C22×C7C22×#4111yescharacteristic
386C22×C7C22×#4111yescharacteristic
387C22×C7C2×2−1+411yescharacteristic
388C22×C7C2×2−1+411yescharacteristic
389C4×C7—24no—
390C4×C7—24no—
391C4×C7—24no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7C2×#12118yes—
395Dic7—28no—
396Dic7—28no—
397Dic7—44no—
398Dic7—44no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24C2×Dic1414yes—
406C24C23×D712yes—
407C24C23×D711yescharacteristic
408C2×Q8—716no—
409C2×Q8—1432no—
410C2×Q8—148no—
411C2×Q8—148no—
412C2×Q8—148no—
413C2×Q8—148no—
414C2×Q8—148no—
415C2×Q8—2816no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—416no—
432C22×C4—78no—
433C22×C4—78no—
434C22×C4—74no—
435C22×C4—74no—
436C22×C4—74no—
437C22×C4—74no—
438C22×C4—144no—
439C22×C4—144no—
440C22×C4—144no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—144no—
444C22×C4—144no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—144no—
448C22×C4C2×Dic1414yes—
449C22×C4C2×Dic1414yes—
450C22×C4#115714yes—
451C22×C4#115714yes—
452C22×C4D4×D714yes—
453C22×C4D4×D714yes—
454C22×C4C23×D712yes—
455C22×C4C23×D712yes—
456C22×C4C23×D712yes—
457C22×C4C23×D712yes—
458C22×C4C23×D712yes—
459C22×C4C23×D712yes—
460C4⋊C4—716no—
461C4⋊C4—1416no—
462C4⋊C4—1416no—
463C4⋊C4—1416no—
464C4⋊C4—1416no—
465C4⋊C4—1416no—
466C4⋊C4—1416no—
467C4⋊C4—148no—
468C22⋊C4—28no—
469C22⋊C4—1416no—
470C22⋊C4—1416no—
471C22⋊C4C2×Dic14116yescomplemented
472C42—28no—
473C42—28no—
474C42—1416no—
475C2×C7—44no—
476C2×C7#242918yescomplemented
477C2×C7C22×#12111yescharacteristic
478C2×C7C2×#53311yescharacteristic
479C2×C7C2×#54111yescharacteristic
480C2×C7C2×#54711yescharacteristic
481C2×C7C2×#54811yescharacteristic
482C2×C7C2×D4×Q811yescharacteristic
483C2×C7C2×D4×Q811yescharacteristic
484C23—28no—
485C23—28no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23C22×Dic1414yes—
498C23C22×Dic1412yes—
499C23C2×#115712yes—
500C23C2×#115712yes—
501C23C2×D4×D712yes—
502C23C2×D4×D712yes—
503C23#655412yes—
504C23#655412yes—
505C23C24×D711yescharacteristic
506Q8—1432no—
507Q8—2816no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—416no—
526C2×C4—78no—
527C2×C4—78no—
528C2×C4—1416no—
529C2×C4—1416no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—288no—
541C2×C4—288no—
542C2×C4#6447116yes—
543C2×C4C22×Dic1414yes—
544C2×C4C22×Dic1414yes—
545C2×C4C2×#115714yes—
546C2×C4C2×#115714yes—
547C2×C4C2×D4×D714yes—
548C2×C4C2×D4×D714yes—
549C7C2×#242911yescharacteristic, complemented, Sylow
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—416no—
558C22#9016114yes—
559C22C2×#644714yes—
560C22#9020914yes—
561C22D4×Dic1414yes—
562C22D4×Dic1414yes—
563C22#9024214yes—
564C22#9024314yes—
565C22C23×Dic1411yescharacteristic
566C22C22×#115711yescharacteristic
567C22C22×#115711yescharacteristic
568C22C22×D4×D711yescharacteristic
569C22C22×D4×D711yescharacteristic
570C22C2×#655411yescharacteristic
571C22C2×#655411yescharacteristic
572C4—24no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—148no—
578C4—148no—
579C4—284no—
580C4—284no—
581C4C2×#644718yes—
582C2—44no—
583C2#1173093318yesdirect product factor
584C2C2×#9016111yescharacteristic
585C2C22×#644711yescharacteristic
586C2C2×#9020911yescharacteristic
587C2C2×#9024311yescharacteristic
588C2C2×D4×Dic1411yescharacteristic
589C2C2×D4×Dic1411yescharacteristic
590C2C2×#9024211yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C4×C7×C22⋊C4C2
socleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×C4×C22⋊C4D7
hypocenterC7C2×#2429
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×#6432C2semidirect
4. C22×C4×Dic14C2semidirect
8. C2×#89701C2semidirect
9. C2×#89700C2semidirect
10. #11730933C2direct (N × H)
11. C2×#89429C2semidirect
12. C2×#89419C2semidirect
36. C2×#6432C22semidirect
38. C2×C4×Dic14C22semidirect
47. #89701C22semidirect
48. #89700C22semidirect
49. #89429C22semidirect
50. #89419C22semidirect
76. C2×C7×C22⋊C4Q8semidirect
188. C7×C22⋊C4C2×Q8semidirect
353. C2×C22⋊C4Dic14semidirect
471. C22⋊C4C2×Dic14semidirect
476. C2×C7#2429semidirect
549. C7C2×#2429semidirect
583. C2#11730933direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C22×#64322
C22×C4×Dic142
C2×C4×C7×C22⋊C42
C2×#897202
C2×#897232
C2×#897012
C2×#897002
#117309332
C2×#894292
C2×#894192
C2×#24297