Finite Group Databasebeta

← #10694894

Subgroups — #10694894

Order 768.

496 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#5051C211yesmaximal, characteristic
2C3×#1472C211yesmaximal, characteristic
3#71688C211yesmaximal, characteristic
4#69982C211yesmaximal, characteristic
5#69980C211yesmaximal, characteristic
6#69947C211yesmaximal, characteristic
7#69923C211yesmaximal, characteristic
8#9814—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#216C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×#835C411yescharacteristic, complemented
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#822—21no—
19C2×C8×Dic3—21no—
20#5051—21no—
21#5051—21no—
22#5051—21no—
23#5051—21no—
24C2×#810C411yescharacteristic, complemented
25C2×#807C411yescharacteristic, complemented
26C3×#324—22no—
27#4558—21no—
28#4550—21no—
29#4538—22no—
30#4469—22no—
31#4467—22no—
32C2×#467—31no—
33#1810—31no—
34C8×C22⋊C4—31no—
35#1508—31no—
36#1508—31no—
37#1472—31no—
38#1472D311yescharacteristic
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×Dic6C2×C411yescharacteristic
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#216—21no—
54C2×#216—21no—
55C2×#216—42no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C22×C3⋊C8—42no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68C22×C3⋊C8D411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75#835—21no—
76#835—21no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#822—42no—
81C8×Dic3—42no—
82#810—21no—
83#810—21no—
84#807—21no—
85#807—21no—
86C22×Q16—31no—
87C23×C8—31no—
88C2×C22⋊Q8—31no—
89C2×C22⋊Q8C2×D311yescharacteristic
90#467—61no—
91#467—61no—
92#467—61no—
93#467—61no—
94C2×#108—61no—
95C2×#107—31no—
96C2×#103—31no—
97C2×#103—31no—
98#407—31no—
99#407C2×D311yescharacteristic
100C2×C22⋊C8C2×D311yescharacteristic
101C2×C4×C8—61no—
102C4×C22⋊C4—31no—
103#341—61no—
104#336—61no—
105#328—62no—
106#328—62no—
107#324—22no—
108#324—62no—
109C24×C3#4111yescharacteristic
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—24no—
144#217—41no—
145#217—41no—
146#216—42no—
147#216—42no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8C22⋊C411yescharacteristic
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8C22⋊C411yescharacteristic
167C22×Q8—31no—
168C22×Q8C4×D311yescharacteristic
169C23×C4C22×D311yescharacteristic
170C2×Q16—61no—
171C2×Q16—61no—
172C2×Q16—122no—
173C2×M16—21no—
174C2×M16—61no—
175C22×C8—21no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—31no—
179C22×C8—31no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—122no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—61no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4C4×D311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C42—61no—
199#108—122no—
200#107—61no—
201#107—61no—
202#103—61no—
203#103—61no—
204#103—61no—
205#103—61no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8#6712yes—
212C4×C8—122no—
213#95—32no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3#12411yescharacteristic
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—81no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×Dic3—24no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—81no—
247Dic6—41no—
248Dic6—41no—
249Dic6—81no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—21no—
253C3⋊C8—21no—
254C3⋊C8—21no—
255C3⋊C8—81no—
256C3⋊C8#9912yes—
257C24#23511yescharacteristic
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—121no—
265C2×Q8—121no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4C2×C4×D311yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—122no—
280Q16—122no—
281M16—41no—
282M16—41no—
283M16—121no—
284M16—121no—
285C2×C8—22no—
286C2×C8—31no—
287C2×C8—31no—
288C2×C8—31no—
289C2×C8—31no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—121no—
297C2×C8—121no—
298C2×C8—121no—
299C2×C8—121no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—64no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—121no—
314C42—121no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3#39011yescharacteristic
324C22×C3#44312yes—
325C22×C3#47312yes—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4C2×#9911yescharacteristic
332C3×C4#41011yescharacteristic
333Dic3—44no—
334Dic3—81no—
335Dic3—81no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23#88411yescharacteristic
340C23#88511yescharacteristic
341C23C4×#6711yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—81no—
345Q8—121no—
346Q8—121no—
347Q8—241no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—64no—
361C2×C4—81no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—241no—
368C2×C4D3×C22⋊C411yescharacteristic
369C2×C4#67⋊C411yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#94011yescharacteristic
372C8—32no—
373C8—42no—
374C8—61no—
375C8—61no—
376C8—61no—
377C8—81no—
378C8—241no—
379C2×C3—22no—
380C2×C3—22no—
381C2×C3—41no—
382C2×C3#196411yescharacteristic
383C2×C3#197411yescharacteristic
384C2×C3#197911yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22#475512yes—
394C22#495711yescharacteristic
395C22#514412yes—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—81no—
401C4—124no—
402C4—241no—
403C4—241no—
404C4#474011yescharacteristic
405C4D3×#9911yescharacteristic
406C3#981411yescharacteristic, complemented, Sylow
407C2—22no—
408C2—22no—
409C2—41no—
410C2#6999911yescharacteristic
411C2#7003011yescharacteristic
412C2#7010811yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#390
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenter#1472D3
hypocenterC3#9814
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
10. C22×#216C4semidirect
15. C2×#835C4semidirect
24. C2×#810C4semidirect
25. C2×#807C4semidirect
406. C3#9814semidirect

Maximal subgroups

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

subgroupindex
C2×#50512
C3×#14722
#716882
#699822
#699802
#699472
#699232
#98143