Finite Group Databasebeta

← #10691264

Subgroups — #10691264

Order 768.

578 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 105 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#74032C211yesmaximal, characteristic
2#73867C211yesmaximal, characteristic, complemented
3#69944C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic
6C3×#1289C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9206—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#886—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×#830C2211yescharacteristic
17#4994—22no—
18#4978—22no—
19#4958—21no—
20#4943—21no—
21Dic3×C22⋊C4—21no—
22#4672C412yescomplemented
23#4661C412yescomplemented
24#4655—22no—
25#4470—22no—
26#4469—22no—
27#4451—22no—
28#2618—31no—
29#2313—31no—
30#1519—31no—
31#1519—31no—
32C2×#324—31no—
33#1289—31no—
34#1289D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12C2×C411yescharacteristic
40C22×C3×C8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×#217—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4C2×C412yes—
61C2×Dic3⋊C4D412yes—
62C2×C4×Dic3—22no—
63C22×C3⋊C8—21no—
64#886—42no—
65#886—42no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95C2×C412yes—
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95—24no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×#123—61no—
77C2×#121—31no—
78C22×C4⋊C4—31no—
79C2×C22⋊C8—31no—
80C2×C22⋊C8C2×D311yescharacteristic
81#398—32no—
82#396—62no—
83#394—32no—
84#387—62no—
85#387—62no—
86#386—61no—
87#381—61no—
88#380—31no—
89#380C2×D311yescharacteristic
90#380C2×D311yescharacteristic
91C4×C22⋊C4—61no—
92#327—62no—
93#327—62no—
94#324—62no—
95C24×C3C2×D411yescharacteristic
96C23×D3—21no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—81no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—41no—
107C2×#67—41no—
108C2×#67—41no—
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—41no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—81no—
123C22×Dic3C22⋊C412yes—
124C22×Dic3C22⋊C412yes—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—81no—
128C2×C3×C8—41no—
129C2×C3×C8—81no—
130C3×C4⋊C4—81no—
131C3×C4⋊C4—81no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136#217—41no—
137#217—41no—
138(C2×D3)⋊C4—44no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—44no—
145C4⋊Dic3—81no—
146Dic3⋊C4—22no—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149Dic3⋊C4—22no—
150C4×Dic3—82no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—81no—
153C22×D4—31no—
154C22×D4—61no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4C22×D311yescharacteristic
158C22×C8—21no—
159C22×C8—61no—
160#123—122no—
161#123—122no—
162#121—61no—
163#121—61no—
164#121—61no—
165#121—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—61no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C42—62no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188#95—32no—
189#95—62no—
190#95—62no—
191#95—64no—
192#95C4×D312yes—
193#95D1212yes—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3C2×C22⋊C411yescharacteristic
197C23×C3#12011yescharacteristic
198C23×C3#12311yescharacteristic
199C23×C3#13511yescharacteristic
200C23×C3#13511yescharacteristic
201C23×C3AGL(1,ℤ/8)11yescharacteristic
202C23×C3AGL(1,ℤ/8)11yescharacteristic
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224#67—81no—
225#67—81no—
226#67—81no—
227#67—81no—
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—81no—
245D12—81no—
246D12—81no—
247C3×C8—81no—
248C3⋊C8—81no—
249C24—61no—
250C24D3×D411yescharacteristic
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—241no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—81no—
275C22×C4—241no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4C2×D1211yescharacteristic
278C22×C4#23511yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23911yescharacteristic
282C2×C8—41no—
283C2×C8—81no—
284C2×C8—121no—
285C2×C8—241no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—124no—
295C4⋊C4—241no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—124no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—242no—
306C42—242no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3#41611yescharacteristic
319C22×C3#41811yescharacteristic
320C22×C3#44911yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#48211yescharacteristic
323C22×C3#48211yescharacteristic
324C2×D3—82no—
325C2×D3—82no—
326C2×D3—81no—
327C2×D3—81no—
328C2×D3—81no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—82no—
334C3×C4—82no—
335Dic3—42no—
336Dic3—42no—
337Dic3—82no—
338Dic3—81no—
339C23—41no—
340C23—41no—
341C23—122no—
342C23—122no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23D3×C22⋊C411yescharacteristic
347C23#88211yescharacteristic
348C23#89411yescharacteristic
349C23#91111yescharacteristic
350C23#91411yescharacteristic
351C23#91611yescharacteristic
352C23#91911yescharacteristic
353D4—241no—
354D4—241no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4—241no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—241no—
388C2×C4C4×D1211yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4#89311yescharacteristic
391C2×C4#89511yescharacteristic
392C8—81no—
393C8—241no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3#187511yescharacteristic
401C2×C3#191211yescharacteristic
402C2×C3#191311yescharacteristic
403D3—161no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—242no—
415C22—242no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22#466511yescharacteristic
420C22#477011yescharacteristic
421C22#478111yescharacteristic
422C22#480411yescharacteristic
423C22#481011yescharacteristic
424C22#486611yescharacteristic
425C22#488511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—82no—
432C4—122no—
433C4—122no—
434C4—242no—
435C4—241no—
436C3#920611yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—481no—
444C2#6870811yescharacteristic
445C2#6888111yescharacteristic
446C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9206
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #73867C2semidirect
4. C2×#4469C2semidirect
6. C3×#1289C2semidirect
7. #67625C2semidirect
22. #4672C4semidirect
23. #4661C4semidirect
34. #1289D3semidirect
436. C3#9206semidirect

Maximal subgroups

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

subgroupindex
#740322
#738672
#699442
C2×#44692
#695092
C3×#12892
#676252
#92063