Finite Group Databasebeta

← #10692366

Subgroups — #10692366

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#82735C211yesmaximal, characteristic
2#73454C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic, complemented
4#71701C211yesmaximal, characteristic, complemented
5#69991C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9364—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#942C412yescomplemented
12C2×C4×Dic6—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#825—22no—
19#5050—21no—
20#5047—22no—
21#5046—21no—
22#4966C412yescomplemented
23#4957—22no—
24#4939—21no—
25#4667—21no—
26#4666—21no—
27C3×#328—21no—
28C3×#328—21no—
29#4526—22no—
30#3007—31no—
31#2341—31no—
32#1996—31no—
33#1789—31no—
34#1508D311yescharacteristic, complemented
35#1305—31no—
36#1297—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—44no—
43C2×#235C2×C412yes—
44C2×#235D412yes—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C22×Dic6D411yescharacteristic
48C2×C3×M16—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#217D412yes—
60C2×#216—22no—
61#942—44no—
62C2×#215—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×Dic3⋊C4—22no—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C2×#208C2×C412yes—
70C4×Dic6—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95D411yescharacteristic
74C3×#95D411yescharacteristic
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#825—44no—
80#810—22no—
81#809—22no—
82C22×#41—31no—
83C2×#137—32no—
84C2×C22⋊Q8C2×D311yescharacteristic
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—31no—
98C4×C22⋊C4—61no—
99C4×C22⋊C4C2×D311yescharacteristic
100#342—62no—
101#328—21no—
102#328—21no—
103C24×C3C2×D411yescharacteristic
104C23×D3—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—42no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—42no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—22no—
120C2×#67—22no—
121C2×#67—44no—
122C22×Dic3—21no—
123C22×Dic3—22no—
124C22×Dic3—41no—
125#235—24no—
126#235—24no—
127#235—42no—
128#235—42no—
129#235—44no—
130#235—44no—
131C2×D12—21no—
132C2×D12—42no—
133C2×D12C22⋊C412yes—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×C4×D3—44no—
137C2×Dic6—21no—
138C2×Dic6—24no—
139C2×Dic6C22⋊C412yes—
140C3×M16—42no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—42no—
150#217—22no—
151#217—41no—
152#217—42no—
153#216—44no—
154#215—44no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—42no—
158Dic3⋊C4—44no—
159C4×Dic3—44no—
160C4×Dic3—42no—
161#208—42no—
162C2×C3⋊C8—22no—
163C2×#41—32no—
164C2×#41—32no—
165C2×#41—61no—
166C2×#41—61no—
167C2×#41—124no—
168C22×Q8—31no—
169C22×Q8C4×D311yescharacteristic
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—61no—
173C23×C4C22×D311yescharacteristic
174#137—124no—
175C2×Q16—62no—
176C2×SD16—62no—
177C2×M16—21no—
178C2×M16—32no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—124no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4C4×D311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42—21no—
195C2×C42—62no—
196C2×C42—61no—
197#104—124no—
198#103—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—61no—
205#95—61no—
206#95#6711yescharacteristic
207#95#6711yescharacteristic
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—
220C3×Q8—42no—
221C3×Q8—81no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242#67—44no—
243#67—44no—
244C2×Dic3—24no—
245C2×Dic3—22no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252D12—22no—
253D12—22no—
254D12—42no—
255C4×D3—44no—
256C4×D3—44no—
257Dic6—24no—
258Dic6—22no—
259Dic6#9914yes—
260C3×C8—81no—
261C3⋊C8—42no—
262C24—61no—
263C24D3×D411yescharacteristic
264#41—64no—
265#41—64no—
266#41—122no—
267#41—122no—
268#41—124no—
269#41—124no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—32no—
273C2×Q8—42no—
274C2×Q8—64no—
275C2×Q8—61no—
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—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—124no—
292C22×C4—121no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299Q16—124no—
300SD16—124no—
301M16—42no—
302M16—122no—
303C2×C8—41no—
304C2×C8—62no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—122no—
310C4⋊C4—124no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—124no—
316C22⋊C4—121no—
317C22⋊C4—122no—
318C22⋊C4—242no—
319C42—42no—
320C42—122no—
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—81no—
346C3×C4—81no—
347C3×C4C2×#9912yes—
348Dic3—24no—
349Dic3—82no—
350Dic3—82no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—62no—
356C23—122no—
357C23—122no—
358C23—121no—
359C23—121no—
360C23#67⋊C411yescharacteristic
361C23#88411yescharacteristic
362C23#93711yescharacteristic
363Q8—34no—
364Q8—42no—
365Q8—64no—
366Q8—62no—
367Q8—81no—
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—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—64no—
388C2×C4—62no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4D3×C22⋊C411yescharacteristic
400C2×C4#88311yescharacteristic
401C2×C4C4×#6711yescharacteristic
402C2×C4#93811yescharacteristic
403C8—81no—
404C8—122no—
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#495611yescharacteristic
429C22#520211yescharacteristic
430C22#520511yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—64no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—242no—
440C4—242no—
441C4D3×#9912yes—
442C3#936411yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—242no—
448C2#7003611yescharacteristic
449C2#7034811yescharacteristic
450C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9364
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
3. C3×#1508C2semidirect
4. #71701C2semidirect
6. #67685C2semidirect
7. #67618C2semidirect
11. C2×#942C4semidirect
22. #4966C4semidirect
34. #1508D3semidirect
442. C3#9364semidirect

Maximal subgroups

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

subgroupindex
#827352
#734542
C3×#15082
#717012
#699912
#676852
#676182
#93643