Finite Group Databasebeta

← #34026392

Subgroups — #34026392

Order 18688.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626440C212yesmaximal, complemented
2#22626121C214yesmaximal, complemented
3#22626120C212yesmaximal, complemented
4#22625793C212yesmaximal
5#22625791C211yesmaximal, characteristic
6#22625678C212yesmaximal
7C4×#14556649C211yesmaximal, characteristic
8C2×#17989295C211yesmaximal, characteristic
9C2×#14556649C2211yescharacteristic
10C2×#14556649C2212yes—
11C2×#14556649C2211yescharacteristic
12#17989538—22no—
13#17989536—24no—
14#17989529C2214yes—
15#17989529C2212yes—
16C4⋊C4×#64408—22no—
17C4×C22⋊#64408C2212yes—
18C4×C22⋊#64408C2214yes—
19#17989465C2214yes—
20#17989465C2214yes—
21#17989465C2212yes—
22#17989465C2212yes—
23#17989295—22no—
24#17989295—24no—
25#17989295—22no—
26C2×#14556590C411yescharacteristic, complemented
27C2×#14556581C412yescomplemented
28C2×#14556571C2212yes—
29C2×#14556568C412yescomplemented
30C2×#14556567C2211yescharacteristic
31C2×Q8×Dic73C411yescharacteristic, complemented
32C2×C4×C22⋊D73C2211yescharacteristic
33C2×#14556551C2212yes—
34C22×C4⋊Dic73C411yescharacteristic, complemented
35C2×C4⋊C4×D73C2211yescharacteristic
36C2×C22⋊Q8×C73C411yescharacteristic, complemented
37C2×C4⋊#64408—22no—
38C2×C4⋊#64408—24no—
39C2×C22⋊#64408C2312yes—
40C2×C22⋊#64408C2314yes—
41C2×C22⋊#64408C2312yes—
42C2×C4×#64408—22no—
43C2×C4×#64408—24no—
44#14556649—22no—
45#14556649—24no—
46#14556649Q814yescomplemented
47(C2×#64408)⋊C4—22no—
48(C2×#64408)⋊C4—24no—
49(C2×#64408)⋊C4—24no—
50(C2×#64408)⋊C4—22no—
51C22×Dic146C2×C412yes—
52C22×C22⋊D73C2311yescharacteristic
53C23×Dic73C2×C412yes—
54C22×C4×D73C2311yescharacteristic
55C22×C4×D73C2312yes—
56#14556590—22no—
57#14556590—22no—
58#14556581—24no—
59#14556581—24no—
60#14556571—24no—
61#14556571—24no—
62#14556568—24no—
63#14556568—24no—
64#14556567—22no—
65#14556567—22no—
66Q8×Dic73—22no—
67Q8×Dic73—21no—
68Q8×Dic73—21no—
69C4×C22⋊D73—22no—
70C4×C22⋊D73—22no—
71#14556551—24no—
72#14556551—24no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2×C411yescharacteristic
77C2×C4⋊Dic73C2×C412yes—
78C2×C4⋊Dic73C2×C412yes—
79C2×C4⋊Dic73C2×C411yescharacteristic
80C2×#11924007C2×C412yes—
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C412yes—
83C2×#11924007C2×C412yes—
84C4⋊C4×D73D412yes—
85C4⋊C4×D73D414yes—
86C4⋊C4×D73D412yes—
87C2×C22⋊Dic73C2×C411yescharacteristic
88C2×#11924005C2312yes—
89C2×#11924005C2311yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C2×C4×Dic73C2×C412yes—
92C22×Q8×C73C2×C411yescharacteristic
93C23×C4×C73C2×C411yescharacteristic
94C22⋊Q8×C73—22no—
95C22⋊Q8×C73—22no—
96C2×C4⋊C4×C73C2×C412yes—
97C2×C4⋊C4×C73C2×C411yescharacteristic
98C2×C22⋊C4×C73C2×C412yes—
99C22×#64408—22no—
100C22×#64408—22no—
101C22×#64408—24no—
102C4⋊#64408—44no—
103C4⋊#64408—84no—
104C22⋊#64408—22no—
105C22⋊#64408—22no—
106C22⋊#64408—24no—
107C22⋊#64408—22no—
108C22⋊#64408—24no—
109C22⋊#64408C2×Q814yes—
110C4×#64408—44no—
111C4×#64408—84no—
112C23×D73C2411yescharacteristic
113C2×Dic146—24no—
114C2×Dic146—82no—
115C2×C22⋊D73—22no—
116C2×C22⋊D73—22no—
117C2×C22⋊D73C2×Q812yes—
118C2×C22⋊D73#4112yes—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—82no—
122C22×Dic73C22×C411yescharacteristic
123C22×Dic73C22×C411yescharacteristic
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C412yes—
126C22×Dic73C22×C411yescharacteristic
127C2×C4×D73—42no—
128C2×C4×D73—82no—
129C2×C4×D73C2×D412yes—
130C2×C4×D73C2×D414yes—
131C2×C4×D73C2×D414yes—
132C2×C4×D73C2×D412yes—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—22no—
136C4⋊Dic73—22no—
137C4⋊Dic73—22no—
138C4⋊Dic73—22no—
139C4⋊Dic73C22⋊C412yes—
140C4⋊Dic73C22⋊C411yescharacteristic
141C4⋊Dic73C22⋊C411yescharacteristic
142#11924007—24no—
143#11924007—22no—
144#11924007—24no—
145#11924007C22⋊C414yes—
146#11924007C22⋊C414yes—
147C22⋊Dic73—22no—
148#11924005—24no—
149#11924005—22no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—22no—
153C4×Dic73—22no—
154C24×C73C22×C411yescharacteristic
155C2×Q8×C73—21no—
156C2×Q8×C73—21no—
157C2×Q8×C73—42no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—42no—
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C22×C412yes—
163C22×C4×C73C22×C411yescharacteristic
164C22×C4×C73C22×C412yes—
165C4⋊C4×C73—24no—
166C4⋊C4×C73C22⋊C412yes—
167C4⋊C4×C73C22⋊C412yes—
168C22⋊C4×C73—24no—
169C23×C73—21no—
170C23×C73—21no—
171C23×C73—41no—
172C23×C73—41no—
173C23×C73C4×Q812yes—
174C23×C73C23×C411yescharacteristic
175C22×D73—41no—
176C22×D73—41no—
177C22×D73C22×D412yes—
178C22×D73C22×Q811yescharacteristic
179C22×D73C2×#4111yescharacteristic
180C22×D732−1+412yes—
181C2×#64408—24no—
182C2×#64408—42no—
183C2×#64408—42no—
184C2×#64408—42no—
185C2×#64408—44no—
186C2×#64408—44no—
187C2×#64408—42no—
188C2×#64408—42no—
189C2×#64408—44no—
190Q8×C73—42no—
191Q8×C73—42no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—21no—
195C2×C4×C73—42no—
196C2×C4×C73—42no—
197C2×C4×C73—42no—
198C2×C4×C73—41no—
199C2×C4×C73—42no—
200C2×C4×C73C2×C22⋊C414yes—
201C2×C4×C73C2×C22⋊C411yescharacteristic
202C2×C4×C73C2×C22⋊C411yescharacteristic
203C22⋊D73—24no—
204C22⋊D73—24no—
205C2×Dic73—21no—
206C2×Dic73—21no—
207C2×Dic73—21no—
208C2×Dic73—22no—
209C2×Dic73—41no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—41no—
213C2×Dic73—82no—
214C2×Dic73—82no—
215C2×Dic73C2×C22⋊C412yes—
216C2×Dic73C2×C22⋊C412yes—
217C2×Dic73C2×C22⋊C412yes—
218C2×Dic73C4×Q812yes—
219C4×D73—24no—
220C4×D73—42no—
221C4×D73—44no—
222C4×D73—44no—
223Dic146—84no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—22no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73C22×C22⋊C411yescharacteristic
232C22×C73C2×C4×Q811yescharacteristic
233C22×C73#51911yescharacteristic
234C2×D73—22no—
235C2×D73—22no—
236C2×D73—41no—
237C2×D73—41no—
238C2×D73—42no—
239C2×D73#54812yes—
240C2×D73D4×Q812yes—
241#64408—44no—
242#64408—82no—
243#64408—84no—
244C4×C73—21no—
245C4×C73—24no—
246C4×C73—41no—
247C4×C73—42no—
248C4×C73#41012yes—
249Dic73—22no—
250Dic73—22no—
251Dic73—22no—
252Dic73—22no—
253Dic73—41no—
254Dic73—42no—
255#22513—731nomaximal, Sylow
256C2×C73—21no—
257C2×C73—21no—
258C2×C73—42no—
259C2×C73Q8×C22⋊C411yescharacteristic
260C2×C73C2×#41011yescharacteristic
261C2×C73#288311yescharacteristic
262D73—22no—
263D73—22no—
264C2×#554—731no—
265#2345—732no—
266#1996—732no—
267#1996—734no—
268#1799—732no—
269#1799—731no—
270C4×#99—731no—
271C4×#99—732no—
272C73#2251311yescharacteristic, complemented, Sylow
273#554—1462no—
274#554—1464no—
275#554—1462no—
276C2×C4⋊Q8—731no—
277C2×#125—732no—
278C2×C22⋊Q8—731no—
279C2×C22⋊Q8—732no—
280C2×C22⋊Q8—732no—
281C2×C22⋊Q8#6440811yescharacteristic, complemented
282C2×C4×Q8—731no—
283C2×C4×D4—731no—
284C2×C4×D4—732no—
285C22×C4⋊C4—731no—
286C22×C4⋊C4—731no—
287C2×#99—731no—
288C2×#99—731no—
289C2×#99—732no—
290#389—1462no—
291#387—1464no—
292#380—734no—
293#380—732no—
294C4×C4⋊C4—1462no—
295C4×C22⋊C4—732no—
296C4×C22⋊C4—734no—
297#342—734no—
298#342—732no—
299#342—734no—
300#342—732no—
301C22×Q8—732no—
302C22×Q8C2×#6440811yescharacteristic
303C22×D4—731no—
304C23×C4—731no—
305C23×C4—732no—
306C23×C4—732no—
307C23×C4C2×#6440811yescharacteristic
308C4⋊Q8—1462no—
309C4⋊Q8—1462no—
310#125—1464no—
311#125—1464no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—1464no—
315C22⋊Q8—1464no—
316C22⋊Q8—1464no—
317C22⋊Q8—1464no—
318C22⋊Q8—1462no—
319C22⋊Q8—1462no—
320C4×Q8—1462no—
321C4×Q8—1461no—
322C4×Q8—1461no—
323C4×D4—1464no—
324C4×D4—1464no—
325C4×D4—1462no—
326C4×D4—1462no—
327C2×C4⋊C4—731no—
328C2×C4⋊C4—732no—
329C2×C4⋊C4—732no—
330C2×C4⋊C4—734no—
331C2×C4⋊C4—732no—
332C2×C4⋊C4—732no—
333C2×C4⋊C4—731no—
334C2×C4⋊C4—731no—
335C2×C4⋊C4—732no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—732no—
338C2×C4⋊C4—1462no—
339C2×C4⋊C4—1464no—
340C2×C4⋊C4—1462no—
341C2×C4⋊C4—1461no—
342C2×C4⋊C4—1461no—
343C2×C4⋊C4C2×#6440812yes—
344C2×C4⋊C4C2×#6440811yescharacteristic
345C2×C22⋊C4—732no—
346C2×C22⋊C4—731no—
347C2×C22⋊C4—731no—
348C2×C22⋊C4—732no—
349C2×C22⋊C4—732no—
350C2×C22⋊C4—734no—
351C2×C22⋊C4C2×#6440812yes—
352C2×C42—732no—
353C2×C42—731no—
354C2×C42—1462no—
355C2×C42—1464no—
356#99—734no—
357#99—1462no—
358#99—1464no—
359#95—1462no—
360#95—1464no—
361#95—1464no—
362#95—1462no—
363C24—731no—
364C24C22×#6440811yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—42no—
368C2×Q8—5842no—
369C2×Q8—1464no—
370C2×D4—732no—
371C2×D4—732no—
372C2×D4—1462no—
373C2×D4—1462no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—2922no—
378C22×C4—5842no—
379C22×C4—5842no—
380C22×C4—732no—
381C22×C4—731no—
382C22×C4—732no—
383C22×C4—734no—
384C22×C4—734no—
385C22×C4—731no—
386C22×C4—732no—
387C22×C4—731no—
388C22×C4—732no—
389C22×C4—1462no—
390C22×C4—1464no—
391C22×C4—1462no—
392C22×C4—1462no—
393C22×C4—1462no—
394C22×C4C22×#6440811yescharacteristic
395C22×C4C22×#6440812yes—
396C22×C4C22×#6440811yescharacteristic
397C22×C4C22×#6440812yes—
398C4⋊C4—24no—
399C4⋊C4—2924no—
400C4⋊C4—5844no—
401C4⋊C4—734no—
402C4⋊C4—734no—
403C4⋊C4—732no—
404C4⋊C4—731no—
405C4⋊C4—731no—
406C4⋊C4—1462no—
407C4⋊C4—1464no—
408C4⋊C4—1462no—
409C4⋊C4—1462no—
410C4⋊C4—1462no—
411C4⋊C4—1462no—
412C4⋊C4—1464no—
413C4⋊C4—1461no—
414C4⋊C4—1461no—
415C4⋊C4C22⋊#6440812yes—
416C4⋊C4C22⋊#6440812yes—
417C22⋊C4—24no—
418C22⋊C4—734no—
419C22⋊C4—1464no—
420C22⋊C4—1462no—
421C22⋊C4—1462no—
422C22⋊C4—1464no—
423C22⋊C4—1462no—
424C22⋊C4—1464no—
425C22⋊C4—1462no—
426C22⋊C4—1462no—
427C42—2924no—
428C42—5844no—
429C42—1462no—
430C42—1462no—
431C42—1461no—
432C42—1461no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—2921no—
438C23—2921no—
439C23—731no—
440C23—731no—
441C23—732no—
442C23—732no—
443C23Q8×#6440812yes—
444C23C23×#6440811yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—5844no—
448D4—1464no—
449D4—1464no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—2924no—
459C2×C4—2924no—
460C2×C4—2922no—
461C2×C4—2922no—
462C2×C4—2922no—
463C2×C4—2922no—
464C2×C4—2921no—
465C2×C4—2922no—
466C2×C4—2924no—
467C2×C4—2922no—
468C2×C4—2924no—
469C2×C4—2924no—
470C2×C4—2922no—
471C2×C4—2921no—
472C2×C4—2922no—
473C2×C4—5842no—
474C2×C4—5842no—
475C2×C4—732no—
476C2×C4—732no—
477C2×C4—732no—
478C2×C4—732no—
479C2×C4—1464no—
480C2×C4—1461no—
481C2×C4—1462no—
482C2×C4—1461no—
483C2×C4—1461no—
484C2×C4—1464no—
485C2×C4C2×C22⋊#6440811yescharacteristic
486C2×C4C2×C22⋊#6440814yes—
487C2×C4C2×C22⋊#6440811yescharacteristic
488C22—21no—
489C22—21no—
490C22—22no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—2921no—
496C22—2921no—
497C22—2922no—
498C22—732no—
499C22—732no—
500C22—1462no—
501C22—1462no—
502C22C22×C22⋊#6440811yescharacteristic
503C22C2×Q8×#6440811yescharacteristic
504C22#1798965511yescharacteristic
505C4—21no—
506C4—24no—
507C4—41no—
508C4—42no—
509C4—2924no—
510C4—2922no—
511C4—2921no—
512C4—5842no—
513C4—5844no—
514C4—1462no—
515C4—1462no—
516C4—1462no—
517C4—1462no—
518C4#1798956712yes—
519C2—21no—
520C2—21no—
521C2—42no—
522C2—1462no—
523C2—1462no—
524C2Q8×C22⋊#6440811yescharacteristic
525C2C2×#1798956711yescharacteristic
526C2#2262653111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×C22⋊D73C23
cosocleC23×D73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#22513
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626440C2semidirect
2. #22626121C2semidirect
3. #22626120C2semidirect
26. C2×#14556590C4semidirect
27. C2×#14556581C4semidirect
29. C2×#14556568C4semidirect
31. C2×Q8×Dic73C4semidirect
34. C22×C4⋊Dic73C4semidirect
36. C2×C22⋊Q8×C73C4semidirect
46. #14556649Q8semidirect
272. C73#22513semidirect
281. C2×C22⋊Q8#64408semidirect

Maximal subgroups

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

subgroupindex
#226264402
#226261212
#226261202
#226257932
#226257912
#226256782
C4×#145566492
C2×#179892952
#2251373