Finite Group Databasebeta

← #10692228

Subgroups — #10692228

Order 768.

520 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 89 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#78642C211yesmaximal, characteristic
2#73886C211yesmaximal, characteristic, complemented
3#69980C211yesmaximal, characteristic, complemented
4#69978C211yesmaximal, characteristic
5#69528C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9344—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5211C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5134—22no—
17#4978—21no—
18#4976—21no—
19C2×#806—21no—
20C2×#806—21no—
21#4945—21no—
22#4944—21no—
23#4738—21no—
24#4736—21no—
25#4731C412yescomplemented
26#4722C412yescomplemented
27#4654—21no—
28#4646—21no—
29#4470—21no—
30#4470—21no—
31#4467—22no—
32#3193—31no—
33#2378—31no—
34#1964—31no—
35#1510—31no—
36#1508—31no—
37#1305D311yescharacteristic, complemented
38#1284—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12C2×C411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60#924C2×C412yes—
61#924D412yes—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#204—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8C2×C412yes—
73C3×#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
82#821—22no—
83#818—22no—
84#816—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8—31no—
88C2×#121—31no—
89#484—32no—
90#480—61no—
91#468—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96#407C2×D311yescharacteristic
97#407C2×D311yescharacteristic
98#391—61no—
99#389—61no—
100#388—61no—
101#388—61no—
102#387—61no—
103#387—61no—
104#381—31no—
105#380C2×D311yescharacteristic
106C4×C22⋊C4—31no—
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—81no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—81no—
136C2×Dic6—21no—
137C2×Dic6—81no—
138C2×Dic6C22⋊C412yes—
139C3×M16—81no—
140C3×M16—81no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C3×C4⋊C4—81no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146#217—22no—
147#217—22no—
148#217—22no—
149#217—41no—
150#217—81no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3C22⋊C412yes—
157Dic3⋊C4—22no—
158Dic3⋊C4—22no—
159Dic3⋊C4—81no—
160Dic3⋊C4—81no—
161C4×Dic3—81no—
162C4×Dic3—81no—
163C4×Dic3—81no—
164#204—82no—
165C22×Q8—31no—
166C22×D4—31no—
167C22×D4—61no—
168C23×C4C22×D311yescharacteristic
169C2×SD16—62no—
170C2×M16—21no—
171C2×M16—21no—
172C22⋊Q8—32no—
173C22⋊Q8—32no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4D1211yescharacteristic
191C2×C22⋊C4D1211yescharacteristic
192C2×C42—61no—
193C2×C42—61no—
194C2×C42—61no—
195#107—62no—
196#103—62no—
197#102—62no—
198#99—122no—
199#99—122no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8C4×D312yes—
204#95—21no—
205#95—31no—
206#95—31no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—81no—
224C22×D3—81no—
225C2×C3×C4—41no—
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—41no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241#67—81no—
242#67—81no—
243#67—81no—
244#67—81no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263D12—42no—
264D12—81no—
265Dic6—42no—
266Dic6—81no—
267C3×C8—42no—
268C3×C8—81no—
269C24—61no—
270C24C2×D1211yescharacteristic
271C2×Q8—32no—
272C2×Q8—61no—
273C2×Q8—241no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—241no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—81no—
291C22×C4—121no—
292C22×C4C2×C4×D311yescharacteristic
293C22×C4#23511yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23911yescharacteristic
298SD16—242no—
299M16—81no—
300M16—81no—
301C2×C8—22no—
302C2×C8—41no—
303C4⋊C4—32no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—81no—
310C4⋊C4—241no—
311C4⋊C4—241no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—241no—
321C42—241no—
322C42—241no—
323C42—241no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—81no—
347C3×C4—81no—
348Dic3—42no—
349Dic3—42no—
350Dic3—81no—
351Dic3—81no—
352Dic3—82no—
353Dic3—81no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—121no—
359C23—121no—
360C23—121no—
361C23—241no—
362C23—241no—
363C23C4×D1211yescharacteristic
364C23#88211yescharacteristic
365C23#89511yescharacteristic
366Q8—122no—
367Q8—241no—
368D4—122no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
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—62no—
387C2×C4—62no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4D3×C22⋊C411yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4#89311yescharacteristic
407C2×C4#89411yescharacteristic
408C8—42no—
409C8—81no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3#188311yescharacteristic
415C2×C3#190511yescharacteristic
416C2×C3#191311yescharacteristic
417D3—161no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—481no—
432C22—481no—
433C22#466511yescharacteristic
434C22#474711yescharacteristic
435C22#475311yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—122no—
443C4—122no—
444C4—241no—
445C4—241no—
446C4—242no—
447C4—241no—
448C3#934411yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—481no—
454C2#6896411yescharacteristic
455C2#6900511yescharacteristic
456C2#6903011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9344
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #73886C2semidirect
3. #69980C2semidirect
6. C3×#1305C2semidirect
7. #67682C2semidirect
25. #4731C4semidirect
26. #4722C4semidirect
37. #1305D3semidirect
448. C3#9344semidirect

Maximal subgroups

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

subgroupindex
#786422
#738862
#699802
#699782
#695282
C3×#13052
#676822
#93443