Finite Group Databasebeta

← #10692370

Subgroups — #10692370

Order 768.

732 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78640C211yesmaximal, characteristic
2#73454C211yesmaximal, characteristic
3#69457C211yesmaximal, characteristic, complemented
4#69454C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9364—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×#909C412yescomplemented
12C2×C4×Dic6—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5212C2211yescharacteristic
17#5207C2211yescharacteristic
18#5133C412yescomplemented
19#4957—22no—
20#4939—21no—
21C2×#805—22no—
22#4739—21no—
23#4733—22no—
24#4731—21no—
25#4667—21no—
26#4666—21no—
27#4527—22no—
28#4452—21no—
29#4452—21no—
30#3007—31no—
31#2341—31no—
32#1996—31no—
33#1789—31no—
34#1508—31no—
35#1305—31no—
36#1297D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—44no—
42C2×#235C2×C412yes—
43C2×#235D412yes—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×Dic6C2×C411yescharacteristic
47C22×Dic6D411yescharacteristic
48C2×C3×M16C2×C412yes—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217D412yes—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—22no—
66C2×#208—21no—
67#909—44no—
68C2×Dic12—22no—
69C2×#204—22no—
70C4×Dic6—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74C3×#95D411yescharacteristic
75C3×#95D411yescharacteristic
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#821—22no—
80#816—22no—
81#805—44no—
82C22×#41—31no—
83C2×#137—32no—
84C2×C22⋊Q8—31no—
85C2×C4×Q8—61no—
86#452—62no—
87#451—61no—
88#450—61no—
89C2×#104—62no—
90#417—32no—
91#407—31no—
92#407C2×D311yescharacteristic
93#390—61no—
94#390—62no—
95#380—31no—
96#380—61no—
97C4×C22⋊C4—61no—
98C4×C22⋊C4C2×D311yescharacteristic
99C4×C22⋊C4C2×D311yescharacteristic
100#342—62no—
101#328—61no—
102#328—61no—
103C24×C3C2×D411yescharacteristic
104C23×D3—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—42no—
108C22×C3×C4—41no—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—22no—
116C2×#67—22no—
117C2×#67—44no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—24no—
123#235—24no—
124#235—42no—
125#235—42no—
126#235—44no—
127#235—44no—
128C2×D12—21no—
129C2×D12—42no—
130C2×D12C22⋊C412yes—
131C2×C4×D3—22no—
132C2×C4×D3—22no—
133C2×C4×D3—44no—
134C2×Dic6—21no—
135C2×Dic6—24no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—42no—
139C2×Dic6C22⋊C412yes—
140C3×M16—42no—
141C2×C3×C8—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—41no—
145C3×C42—42no—
146C3×C42—42no—
147#217—41no—
148#217—41no—
149#217—42no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—82no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—42no—
155Dic3⋊C4—44no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—44no—
159#208—42no—
160C2×C3⋊C8—41no—
161Dic12—44no—
162#204—44no—
163C2×#41—32no—
164C2×#41—32no—
165C2×#41—61no—
166C2×#41—61no—
167C2×#41—124no—
168C22×Q8—31no—
169C22×Q8—31no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—61no—
173C23×C4C22×D311yescharacteristic
174#137—124no—
175C2×Q16—62no—
176C2×SD16—62no—
177C2×M16—61no—
178C2×M16C4×D312yes—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C4×Q8—124no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—62no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—62no—
197#104—124no—
198#103—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—21no—
205#95—61no—
206#95D1211yescharacteristic
207#95D1211yescharacteristic
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C22×D3—22no—
216C22×D3—42no—
217C22×D3—42no—
218C22×D3—41no—
219C22×D3—41no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239#67—44no—
240#67—44no—
241C2×Dic3—24no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250D12—22no—
251D12—22no—
252D12—42no—
253C4×D3—44no—
254C4×D3—44no—
255Dic6—24no—
256Dic6—22no—
257Dic6—42no—
258Dic6—81no—
259Dic6#9914yes—
260C3×C8—42no—
261C3⋊C8—81no—
262C24—61no—
263C24D3×D411yescharacteristic
264#41—64no—
265#41—64no—
266#41—122no—
267#41—122no—
268#41—124no—
269#41—124no—
270C2×Q8—32no—
271C2×Q8—64no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×D4—32no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—61no—
280C2×D4—122no—
281C2×D4—124no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—124no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4#23911yescharacteristic
299Q16—124no—
300SD16—124no—
301M16—42no—
302M16—122no—
303C2×C8—22no—
304C2×C8—121no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—122no—
308C4⋊C4—124no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—22no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—124no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—122no—
318C22⋊C4—242no—
319C42—42no—
320C42—42no—
321C42—124no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3#38611yescharacteristic
333C22×C3#45511yescharacteristic
334C22×C3#45611yescharacteristic
335C2×D3—42no—
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—82no—
339C2×D3—82no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—42no—
344C3×C4—81no—
345C3×C4—82no—
346C3×C4C2×#9912yes—
347Dic3—24no—
348Dic3—82no—
349Dic3—81no—
350Dic3—81no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—62no—
356C23—122no—
357C23—122no—
358C23—121no—
359C23—121no—
360C23#88211yescharacteristic
361C23#88411yescharacteristic
362C23#89311yescharacteristic
363Q8—34no—
364Q8—64no—
365Q8—62no—
366Q8—122no—
367Q8—241no—
368D4—62no—
369D4—62no—
370D4—122no—
371D4—124no—
372D4—124no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—64no—
387C2×C4—62no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—124no—
391C2×C4—124no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4C4×D1211yescharacteristic
400C2×C4D3×C22⋊C411yescharacteristic
401C2×C4#89411yescharacteristic
402C2×C4#89511yescharacteristic
403C8—42no—
404C8—241no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3#188211yescharacteristic
410C2×C3#189811yescharacteristic
411C2×C3#190611yescharacteristic
412D3—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—81no—
422C22—81no—
423C22—122no—
424C22—122no—
425C22—122no—
426C22—242no—
427C22—242no—
428C22#466511yescharacteristic
429C22#482511yescharacteristic
430C22#482611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—64no—
436C4—81no—
437C4—82no—
438C4—242no—
439C4—241no—
440C4—241no—
441C4D3×#9912yes—
442C3#936411yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—242no—
448C2#6877511yescharacteristic
449C2#6877811yescharacteristic
450C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9364
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #69457C2semidirect
5. #69445C2semidirect
6. C3×#1297C2semidirect
7. #67684C2semidirect
11. C2×#909C4semidirect
18. #5133C4semidirect
36. #1297D3semidirect
442. C3#9364semidirect

Maximal subgroups

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

subgroupindex
#786402
#734542
#694572
#694542
#694452
C3×#12972
#676842
#93643