Finite Group Databasebeta

← #34022742

Subgroups — #34022742

Order 18688.

1074 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626611C211yesmaximal, characteristic, complemented
2#22626495C211yesmaximal, characteristic, complemented
3#22626124C212yesmaximal, complemented
4#22626075C212yesmaximal, complemented
5#22625958C212yesmaximal, complemented
6#22625956C212yesmaximal
7#22625684C212yesmaximal, complemented
8C4×#14556651C212yesmaximal, complemented
9C2×#17989205C211yesmaximal, characteristic
10C22×#11924037C2212yes—
11#17989611—24no—
12#17989590C2212yes—
13C2×#14556652C2214yes—
14C2×#14556651C2214yes—
15#17989557C2212yes—
16#17989551—24no—
17C4×#11924037C2212yes—
18C4×#11924037C2212yes—
19C4×#11924037C2212yes—
20#17989467C2214yes—
21#17989467C2214yes—
22#17989205—24no—
23#17989205—24no—
24C2×#14556590C2211yescharacteristic
25C2×#14556586C411yescharacteristic
26C2×#14556584C411yescharacteristic
27C2×#14556578C411yescharacteristic
28C2×#14556577C411yescharacteristic
29C2×#14556577C2211yescharacteristic
30C2×#14556573C411yescharacteristic
31C2×#14556570C411yescharacteristic
32C2×#14556569C411yescharacteristic
33C2×#14556568C2211yescharacteristic
34C2×#14556563C2211yescharacteristic
35C2×#14556556C2211yescharacteristic
36C2×C4×C22⋊D73C2211yescharacteristic
37C2×#14556547C2211yescharacteristic
38C2×#123×C73C411yescharacteristic
39C2×#11924037—28no—
40C2×#11924037—22no—
41C2×#11924037C2314yes—
42C2×#11924037C2314yes—
43C22×#10596833—22no—
44#14556658—22no—
45C4⋊#10596833—22no—
46#14556652—28no—
47#14556651—28no—
48C22⋊#10596833—24no—
49#14556646—22no—
50#14556646—22no—
51#14556646—22no—
52C4×#10596833—22no—
53C4×#10596833—22no—
54C4×#10596833—22no—
55C22×Dic146C2×C411yescharacteristic
56C22×Dic146C2×C411yescharacteristic
57C22×Dic146C2311yescharacteristic
58C22×C22⋊D73C2×C411yescharacteristic
59C22×C22⋊D73C2311yescharacteristic
60C23×Dic73C2311yescharacteristic
61C22×C4×D73C2×C411yescharacteristic
62#14556590—22no—
63#14556590—22no—
64#14556586—24no—
65#14556584—24no—
66#14556578—24no—
67#14556577—24no—
68#14556577—22no—
69#14556577—22no—
70#14556573—24no—
71#14556570—24no—
72#14556569—24no—
73#14556568—22no—
74#14556568—22no—
75#14556563—22no—
76#14556563—22no—
77#14556556—22no—
78#14556556—22no—
79C4×C22⋊D73—22no—
80C4×C22⋊D73—22no—
81#14556547D414yes—
82#14556547D414yes—
83C2×C4⋊Dic73C2×C411yescharacteristic
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×#11924007C2×C411yescharacteristic
89C2×#11924007C2311yescharacteristic
90C2×#11924007C2311yescharacteristic
91C2×C22⋊Dic73C2×C411yescharacteristic
92C2×C22⋊Dic73C2×C411yescharacteristic
93C2×C22⋊Dic73C2×C411yescharacteristic
94C2×C22⋊Dic73C2×C411yescharacteristic
95C2×C22⋊Dic73C2×C411yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×C4×Dic73C2×C411yescharacteristic
101C2×C4×Dic73C2×C411yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C22×D4×C73C2×C411yescharacteristic
105C23×C4×C73C2×C411yescharacteristic
106#123×C73—24no—
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C4⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110C2×C22⋊C4×C73C2×C411yescharacteristic
111C2×C22⋊C4×C73C2×C411yescharacteristic
112#11924037—28no—
113#11924037—24no—
114#11924037—24no—
115#11924037—24no—
116#11924037#4118yes—
117C2×#10596833—28no—
118C2×#10596833—24no—
119C2×#10596833—24no—
120C23×D73C22×C411yescharacteristic
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—24no—
124C2×Dic146—22no—
125C2×Dic146—22no—
126C2×Dic146—81no—
127C2×Dic146—81no—
128C2×C22⋊D73—22no—
129C2×C22⋊D73—21no—
130C2×C22⋊D73—21no—
131C2×C22⋊D73—24no—
132C2×C22⋊D73—81no—
133C22×Dic73—24no—
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C2×D412yes—
141C22×Dic73C2×D412yes—
142C22×Dic73C2411yescharacteristic
143C2×C4×D73—22no—
144C2×C4×D73—81no—
145C4⋊Dic73—22no—
146C4⋊Dic73—22no—
147#11924007—22no—
148#11924007—22no—
149#11924007—21no—
150#11924007—21no—
151#11924007—21no—
152#11924007—21no—
153#11924007C22⋊C414yes—
154#11924007C22⋊C414yes—
155C22⋊Dic73—22no—
156C22⋊Dic73—22no—
157C22⋊Dic73—22no—
158C22⋊Dic73—22no—
159C22⋊Dic73C22⋊C414yes—
160#11924005—22no—
161#11924005—22no—
162#11924005—22no—
163#11924005—22no—
164C4×Dic73—22no—
165C4×Dic73—22no—
166C4×Dic73C2×D412yes—
167C4×Dic73C2×D412yes—
168C4×Dic73C2×D412yes—
169C4×Dic73C2×D412yes—
170C24×C73C22×C411yescharacteristic
171C24×C73C22×C411yescharacteristic
172C2×D4×C73—22no—
173C2×D4×C73—81no—
174C22×C4×C73—22no—
175C22×C4×C73—81no—
176C22×C4×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73C22⋊C414yes—
186C23×C73—22no—
187C23×C73—21no—
188C23×C73—22no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73C2×C22⋊C412yes—
194C23×C73C23×C411yescharacteristic
195C22×D73—21no—
196C22×D73—22no—
197C22×D73—41no—
198C22×D73—41no—
199D4×C73—82no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—22no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—82no—
209C2×C4×C73C2×C22⋊C412yes—
210C2×C4×C73C2×C22⋊C412yes—
211C22⋊D73—42no—
212C22⋊D73—42no—
213C22⋊D73—82no—
214C2×Dic73—22no—
215C2×Dic73—22no—
216C2×Dic73—22no—
217C2×Dic73—24no—
218C2×Dic73—21no—
219C2×Dic73—21no—
220C2×Dic73—21no—
221C2×Dic73—22no—
222C2×Dic73—41no—
223C2×Dic73—41no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73C2×C22⋊C412yes—
227C2×Dic73C2×C22⋊C412yes—
228C2×Dic73C2×C22⋊C412yes—
229C2×Dic73C22×D411yescharacteristic
230C2×Dic73C22×D411yescharacteristic
231C2×Dic73C2×#4114yes—
232C4×D73—82no—
233Dic146—42no—
234Dic146—42no—
235Dic146—82no—
236Dic146—82no—
237#10596833—28no—
238#10596833—44no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—22no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—41no—
245C22×C73—41no—
246C22×C73—41no—
247C22×C73—82no—
248C22×C73—82no—
249C22×C73C22×C22⋊C411yescharacteristic
250C22×C73C4×#4112yes—
251C2×D73—41no—
252C2×D73—41no—
253C2×D73—41no—
254C2×D73—82no—
255C4×C73—22no—
256C4×C73—22no—
257C4×C73—41no—
258C4×C73—41no—
259C4×C73—41no—
260Dic73—22no—
261Dic73—22no—
262Dic73—22no—
263Dic73—41no—
264Dic73—41no—
265Dic73—41no—
266Dic73#52512yes—
267Dic73#53212yes—
268#21427—731nomaximal, Sylow
269C2×C73—22no—
270C2×C73—42no—
271C2×C73—81no—
272C2×C73#229711yescharacteristic
273C2×C73#288811yescharacteristic
274C2×C73#288911yescharacteristic
275D73—81no—
276C2×#542—731no—
277#2921—731no—
278#2873—731no—
279#1997—732no—
280#1967—732no—
281#1901—732no—
282#1900—732no—
283C4×#101—732no—
284C4×#100—732no—
285C73#2142711yescharacteristic, complemented, Sylow
286C22×M16—732no—
287#542—1464no—
288#542—1464no—
289C2×C4⋊Q8—731no—
290C2×#126—731no—
291C2×#126—731no—
292C2×#124—731no—
293C2×#124—731no—
294C2×#124—731no—
295C2×#123—731no—
296C2×#123#6440811yescharacteristic
297C2×C22⋊Q8—731no—
298C2×C22⋊Q8—731no—
299C2×C22⋊Q8—731no—
300C2×#121—731no—
301C2×C4×Q8—731no—
302C2×C4×D4—731no—
303C2×#117—731no—
304#433—1464no—
305#423—732no—
306C2×#101—734no—
307C2×#100—734no—
308#407—732no—
309#405—1464no—
310C4×M16—732no—
311C4×M16—732no—
312C4×M16—732no—
313#343—734no—
314#343—734no—
315C22×Q8—731no—
316C22×Q8—731no—
317C22×Q8—731no—
318C22×D4—731no—
319C22×D4—731no—
320C22×D4C2×#6440811yescharacteristic
321C23×C4—731no—
322C23×C4—731no—
323C23×C4C2×#6440811yescharacteristic
324C2×M16—734no—
325C2×M16—734no—
326C2×M16—1468no—
327C2×M16—1462no—
328C22×C8—1462no—
329C4⋊Q8—1462no—
330C4⋊Q8—1462no—
331#126—1464no—
332#126—1464no—
333#124—1462no—
334#124—1462no—
335#124—1464no—
336#124—1464no—
337#123—24no—
338#123—1464no—
339C22⋊Q8—1462no—
340C22⋊Q8—1462no—
341C22⋊Q8—1464no—
342C22⋊Q8—1464no—
343#121—1462no—
344#121—1462no—
345C4×Q8—1462no—
346C4×Q8—1462no—
347C4×D4—1462no—
348C4×D4—1462no—
349#117—734no—
350#117—734no—
351C2×C4⋊C4—731no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—731no—
354C2×C4⋊C4—731no—
355C2×C4⋊C4—731no—
356C2×C4⋊C4—731no—
357C2×C4⋊C4—731no—
358C2×C4⋊C4—731no—
359C2×C4⋊C4C2×#6440811yescharacteristic
360C2×C4⋊C4C2×#6440811yescharacteristic
361C2×C22⋊C4—731no—
362C2×C22⋊C4—731no—
363C2×C22⋊C4—731no—
364C2×C22⋊C4—731no—
365C2×C22⋊C4—731no—
366C2×C22⋊C4—731no—
367C2×C22⋊C4—731no—
368C2×C22⋊C4—731no—
369C2×C22⋊C4—731no—
370C2×C22⋊C4C2×#6440811yescharacteristic
371C2×C22⋊C4C2×#6440811yescharacteristic
372C2×C22⋊C4C2×#6440811yescharacteristic
373C2×C42—731no—
374C2×C42—731no—
375C2×C42—731no—
376C2×C42—731no—
377C4⋊C8—1462no—
378C4⋊C8—1462no—
379#101—1468no—
380#100—1468no—
381C22⋊C8—1464no—
382#97—1462no—
383#97—1462no—
384#97—1462no—
385C4×C8—1462no—
386C4×C8—1462no—
387C4×C8—1462no—
388C24—731no—
389C24C22×#6440811yescharacteristic
390C24C22×#6440811yescharacteristic
391C2×Q8—5841no—
392C2×Q8—5841no—
393C2×Q8—1464no—
394C2×Q8—1461no—
395C2×Q8—1461no—
396C2×Q8—1462no—
397C2×Q8—1462no—
398C2×D4—22no—
399C2×D4—81no—
400C2×D4—5841no—
401C2×D4—1464no—
402C2×D4—1461no—
403C2×D4—1461no—
404C2×D4—1462no—
405C22×C4—22no—
406C22×C4—81no—
407C22×C4—5841no—
408C22×C4—731no—
409C22×C4—731no—
410C22×C4—731no—
411C22×C4—731no—
412C22×C4—731no—
413C22×C4—731no—
414C22×C4—732no—
415C22×C4—732no—
416C22×C4—731no—
417C22×C4—1462no—
418C22×C4—1464no—
419C22×C4C22×#6440811yescharacteristic
420C22×C4C22×#6440811yescharacteristic
421C22×C4C22×#6440811yescharacteristic
422C22×C4C22×#6440811yescharacteristic
423C22×C4C22×#6440811yescharacteristic
424M16—738no—
425M16—1464no—
426M16—1464no—
427M16—1464no—
428M16—1468no—
429C2×C8—1464no—
430C2×C8—1464no—
431C2×C8—1468no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—734no—
435C4⋊C4—734no—
436C4⋊C4—1461no—
437C4⋊C4—1461no—
438C4⋊C4—1461no—
439C4⋊C4—1461no—
440C4⋊C4—1462no—
441C4⋊C4—1462no—
442C4⋊C4—1462no—
443C4⋊C4—1462no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—734no—
447C22⋊C4—1462no—
448C22⋊C4—1462no—
449C22⋊C4—1462no—
450C22⋊C4—1462no—
451C22⋊C4—1462no—
452C22⋊C4—1462no—
453C22⋊C4—1462no—
454C22⋊C4—1462no—
455C22⋊C4C22⋊#6440814yes—
456C42—732no—
457C42—732no—
458C42—732no—
459C42—732no—
460C42—1462no—
461C42—1462no—
462C23—22no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—2921no—
470C23—2921no—
471C23—1461no—
472C23—1462no—
473C23C2×C22⋊#6440812yes—
474C23C23×#6440811yescharacteristic
475Q8—2922no—
476Q8—2922no—
477Q8—5842no—
478Q8—5842no—
479D4—82no—
480D4—2922no—
481D4—2922no—
482D4—5842no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—2921no—
493C2×C4—2921no—
494C2×C4—2921no—
495C2×C4—2921no—
496C2×C4—5842no—
497C2×C4—731no—
498C2×C4—731no—
499C2×C4—734no—
500C2×C4—732no—
501C2×C4—732no—
502C2×C4—732no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4—1462no—
507C2×C4—1462no—
508C2×C4—1462no—
509C2×C4—1464no—
510C2×C4—1462no—
511C2×C4C2×C22⋊#6440812yes—
512C2×C4C2×C22⋊#6440812yes—
513C8—2924no—
514C8—1468no—
515C22—21no—
516C22—21no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—82no—
524C22—82no—
525C22—2921no—
526C22—2921no—
527C22—2921no—
528C22—5842no—
529C22C22×C22⋊#6440811yescharacteristic
530C22#41×#6440812yes—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—2921no—
537C4—2921no—
538C4—2921no—
539C4—732no—
540C4—732no—
541C4—1462no—
542C4—1462no—
543C4—1462no—
544C2—22no—
545C2—42no—
546C2—81no—
547C2—5841no—
548C2#2262639211yescharacteristic
549C2#2262654111yescharacteristic
550C2#2262654311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#123×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC2×#14556563C22
cosocleC22×Dic73C24
hypercenterC2×#123#64408
hypocenterC73#21427
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #22626611C2semidirect
2. #22626495C2semidirect
3. #22626124C2semidirect
4. #22626075C2semidirect
5. #22625958C2semidirect
7. #22625684C2semidirect
8. C4×#14556651C2semidirect
285. C73#21427semidirect

Maximal subgroups

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

subgroupindex
#226266112
#226264952
#226261242
#226260752
#226259582
#226259562
#226256842
C4×#145566512
C2×#179892052
#2142773