Finite Group Databasebeta

← #10695883

Subgroups — #10695883

Order 768.

698 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 87 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#82742C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic
3C2×#4739C211yesmaximal, characteristic
4C3×#1472C211yesmaximal, characteristic
5#69990C211yesmaximal, characteristic
6#69972C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic
8#9991—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#216—21no—
12C2×#924C2211yescharacteristic
13C22×Dic12—21no—
14C3×#407C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16#5228—21no—
17#5207C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#824—22no—
20C2×#816—21no—
21#5052—22no—
22#5050—22no—
23#5050—21no—
24#5050—22no—
25#5043—21no—
26#4971—21no—
27C2×#810—21no—
28#4739—22no—
29#4739—22no—
30C3×#324—22no—
31#4469—22no—
32#3195—31no—
33C2×#451—31no—
34C2×#451—31no—
35#1784—31no—
36#1472—31no—
37#1472D311yescharacteristic
38#1298—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—44no—
48C22×Dic6D412yes—
49C22×Dic6D411yescharacteristic
50C2×C3×M16—21no—
51C22×C3×C8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#216—24no—
61C2×#216—22no—
62C2×#216—44no—
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×#208D412yes—
70C22×C3⋊C8—21no—
71C2×Dic12—24no—
72C2×Dic12—44no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77#835—22no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830D412yes—
82#824—44no—
83#816—24no—
84#810—24no—
85#810—22no—
86#808—22no—
87C23×Q8—31no—
88C22×Q16—61no—
89C22×Q16—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#483—61no—
93#483—61no—
94#470—62no—
95#451—62no—
96#451—62no—
97#451—61no—
98#451—62no—
99#451—61no—
100#451—62no—
101C2×#103—61no—
102C2×#103—61no—
103C2×#101—62no—
104#407—31no—
105#407C2×D311yescharacteristic
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#324—22no—
109#324—62no—
110C24×C3#4111yescharacteristic
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—42no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—44no—
127C22×Dic3—41no—
128C22×Dic3—41no—
129C2×Dic6—24no—
130C2×Dic6—24no—
131C2×Dic6—22no—
132C2×Dic6—22no—
133C2×Dic6—24no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—44no—
137C2×Dic6—44no—
138C2×Dic6—44no—
139C2×Dic6—42no—
140C2×Dic6—42no—
141C2×Dic6—42no—
142C2×Dic6—41no—
143C2×Dic6—41no—
144C2×Dic6—42no—
145C3×M16—42no—
146C2×C3×C8—22no—
147C2×C3×C8—42no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155#217—41no—
156#217—41no—
157#216—48no—
158#216—82no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—42no—
167Dic12—48no—
168C22×Q8—31no—
169C22×Q8—32no—
170C22×Q8—61no—
171C22×Q8—61no—
172C22×Q8—61no—
173C22×Q8—61no—
174C22×Q8—61no—
175C22×Q8—124no—
176C22×Q8D1211yescharacteristic
177C23×C4—61no—
178C23×C4C22×D311yescharacteristic
179C2×Q16—64no—
180C2×Q16—64no—
181C2×Q16—62no—
182C2×Q16—124no—
183C2×Q16—124no—
184C2×M16—21no—
185C2×M16—32no—
186C22×C8—21no—
187C22×C8—61no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4D1211yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—61no—
202#106—62no—
203#103—64no—
204#103—62no—
205#103—64no—
206#103—62no—
207#101—124no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8#6712yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3#12111yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3#12111yescharacteristic
222C3×Q8—42no—
223C3×Q8—42no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12711yescharacteristic
238C2×Dic3—24no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—81no—
247Dic6—24no—
248Dic6—24no—
249Dic6—42no—
250Dic6—42no—
251Dic6—44no—
252Dic6—44no—
253Dic6—42no—
254Dic6—42no—
255C3×C8—42no—
256C3×C8—81no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C24#23511yescharacteristic
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—42no—
263C2×Q8—64no—
264C2×Q8—64no—
265C2×Q8—64no—
266C2×Q8—62no—
267C2×Q8—61no—
268C2×Q8—62no—
269C2×Q8—61no—
270C2×Q8—122no—
271C2×Q8—122no—
272C2×Q8—122no—
273C2×Q8—122no—
274C2×Q8—124no—
275C2×Q8—124no—
276C2×Q8—124no—
277C2×Q8—121no—
278C2×Q8—121no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—62no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—124no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4C2×#6711yescharacteristic
294Q16—128no—
295Q16—128no—
296Q16—242no—
297M16—42no—
298M16—122no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—122no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—44no—
325C22×C3#39211yescharacteristic
326C22×C3#47312yes—
327C22×C3#47312yes—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4B(4,𝔽2)12yes—
334Dic3—44no—
335Dic3—81no—
336Dic3—81no—
337C23—22no—
338C23—22no—
339C23—41no—
340C23#88411yescharacteristic
341C23#88411yescharacteristic
342C23#93011yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—64no—
346Q8—64no—
347Q8—122no—
348Q8—124no—
349Q8—122no—
350Q8—122no—
351Q8—124no—
352Q8—122no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—64no—
363C2×C4—81no—
364C2×C4—122no—
365C2×C4—124no—
366C2×C4—124no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—241no—
372C2×C4#88211yescharacteristic
373C2×C4#88211yescharacteristic
374C2×C4#93711yescharacteristic
375C2×C4#94011yescharacteristic
376C8—42no—
377C8—81no—
378C8—122no—
379C8—122no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—41no—
383C2×C3#202011yescharacteristic
384C2×C3#202011yescharacteristic
385C2×C3#202111yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22#475512yes—
395C22#496011yescharacteristic
396C22#514312yes—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—81no—
401C4—81no—
402C4—124no—
403C4—241no—
404C4—241no—
405C4#474112yes—
406C3#999111yescharacteristic, complemented, Sylow
407C2—22no—
408C2—22no—
409C2—41no—
410C2#7000811yescharacteristic
411C2#7003711yescharacteristic
412C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#1472D3
hypocenterC3#9991
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
406. C3#9991semidirect

Maximal subgroups

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

subgroupindex
#827422
C2×#50502
C2×#47392
C3×#14722
#699902
#699722
#676802
#99913