Finite Group Databasebeta

← #10696099

Subgroups — #10696099

Order 768.

518 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 79 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#82391C211yesmaximal, characteristic
2#82315C211yesmaximal, characteristic
3#78507C211yesmaximal, characteristic, complemented
4C3×#1490C211yesmaximal, characteristic
5#70837C211yesmaximal, characteristic, complemented
6#70814C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#10023—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5222—21no—
15#5218—21no—
16#5210—22no—
17#5121—21no—
18#5121—21no—
19#5019—21no—
20#5018—22no—
21#5007—21no—
22#4983—21no—
23#4970—21no—
24#4953—21no—
25#4949—21no—
26#4937C2211yescharacteristic
27#4926C2211yescharacteristic
28#4702—21no—
29#4702—21no—
30#4683—22no—
31C3×D4⋊C8—22no—
32#4467—22no—
33#4465—22no—
34#3166—31no—
35#3110—31no—
36#2633—31no—
37#1534—31no—
38#1515—31no—
39#1490D311yescharacteristic
40#1288—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×M16—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52C2×#217—22no—
53C2×#214—22no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97—21no—
75C3×C4×C8D412yes—
76#832—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#818—22no—
83#818—22no—
84#817—22no—
85#817—22no—
86#810—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89#803D412yes—
90#533—31no—
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#499—62no—
96#484—61no—
97#474—61no—
98#461—61no—
99#450—61no—
100#440—62no—
101#432—31no—
102#432C2×D311yescharacteristic
103#428—61no—
104#428—61no—
105C4×M16C2×D311yescharacteristic
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#328—62no—
114D4⋊C8—22no—
115C24×C3—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—82no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C3×M16—24no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#217—44no—
143#217—41no—
144#217—82no—
145#214—44no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—82no—
161C4×Dic3—41no—
162C2×C3⋊C8—22no—
163C22×D4D1211yescharacteristic
164C23×C4—21no—
165C2×SD16—62no—
166C2×M16—21no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182#117—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4D1211yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C42C22×D311yescharacteristic
193#107—62no—
194#107—62no—
195#107—62no—
196#106—62no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200#103—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204#97—21no—
205#97—21no—
206#97—32no—
207C4×C8#6712yes—
208#95—32no—
209#95—62no—
210#95—61no—
211#95—61no—
212C23×C3—22no—
213C23×C3—42no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3#12011yescharacteristic
218C3×D4—22no—
219C3×D4—22no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—82no—
228C2×C3×C4—82no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12711yescharacteristic
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245Dic6—42no—
246Dic6—81no—
247C3×C8—24no—
248C3×C8—81no—
249C3⋊C8—42no—
250C24—21no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—82no—
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266SD16—124no—
267M16—24no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—62no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—242no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—124no—
291C22⋊C4—121no—
292C22⋊C4—242no—
293C42—121no—
294C42#23511yescharacteristic
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—82no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3#39211yescharacteristic
306C22×C3#45612yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—81no—
314Dic3—82no—
315Dic3—81no—
316Dic3—81no—
317Dic3—81no—
318Dic3—81no—
319C23—22no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23#88211yescharacteristic
325Q8—122no—
326Q8—241no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4#88211yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#88411yescharacteristic
351C2×C4#93011yescharacteristic
352C2×C4#93711yescharacteristic
353C2×C4#94011yescharacteristic
354C8—24no—
355C8—81no—
356C8—122no—
357C2×C3—22no—
358C2×C3—82no—
359C2×C3#200511yescharacteristic
360C2×C3#200811yescharacteristic
361C2×C3#201011yescharacteristic
362C22—21no—
363C22—21no—
364C22—44no—
365C22—42no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22#482612yes—
370C22#496011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—242no—
379C4—241no—
380C4—241no—
381C4—241no—
382C4—241no—
383C3#1002311yescharacteristic, complemented, Sylow
384C2—22no—
385C2—82no—
386C2#6999311yescharacteristic
387C2#7033111yescharacteristic
388C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1490C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1490D3
hypocenterC3#10023
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
3. #78507C2semidirect
5. #70837C2semidirect
6. #70814C2semidirect
7. #67666C2semidirect
383. C3#10023semidirect

Maximal subgroups

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

subgroupindex
#823912
#823152
#785072
C3×#14902
#708372
#708142
#676662
#100233