Finite Group Databasebeta

← #10692529

Subgroups — #10692529

Order 768.

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

normal subgroups only

This group has 496 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#83000C211yesmaximal, characteristic
2#78518C211yesmaximal, characteristic, complemented
3C3×#1527C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#70826C211yesmaximal, characteristic, complemented
6#67748C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9389—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#872C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5231C411yescharacteristic
14#5230C411yescharacteristic
15#5223C411yescharacteristic
16#5222C411yescharacteristic
17#5126—21no—
18#5118—21no—
19#4957—21no—
20#4956—21no—
21#4945—21no—
22#4944—21no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4709—22no—
27#4690—22no—
28C3×#329—22no—
29#4540—22no—
30#4469—21no—
31#4468—21no—
32#2921—31no—
33#2610—31no—
34#1527—31no—
35#1527D311yescharacteristic, complemented
36#1464—31no—
37#1300—31no—
38#1300—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×#67—21no—
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4C2×C411yescharacteristic
56C2×(C2×D3)⋊C4D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4C2×C411yescharacteristic
59C2×Dic3⋊C4D411yescharacteristic
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×#208—21no—
63#872—21no—
64#872—21no—
65#872—21no—
66#872—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97D412yescomplemented
71C3×C4×C8—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95C2×C411yescharacteristic
76C3⋊#95C2×C411yescharacteristic
77#830—21no—
78(C2×D3)⋊C8—24no—
79Dic3⋊C8—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803D412yescomplemented
83C4×C3⋊C8—21no—
84C2×#124C2×D311yescharacteristic
85C2×#117—31no—
86#433—62no—
87#433—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90#423—61no—
91#407—61no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#395—31no—
95#394—31no—
96#393—31no—
97#392—31no—
98#390—61no—
99#388—61no—
100#388—61no—
101#386—61no—
102#376—31no—
103#329—22no—
104#329—62no—
105#324—61no—
106#323—61no—
107C24×C3—21no—
108C23×D3C22⋊C411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—81no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—81no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—42no—
120C2×#67—42no—
121C2×#67—81no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C22×Dic3C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C3×C42#4111yescharacteristic
143#217—42no—
144#217—42no—
145#217—81no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—81no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—81no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—42no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×Q8C4×D311yescharacteristic
164C22×D4—61no—
165C22×D4C4×D311yescharacteristic
166C23×C4—31no—
167C2×M16—21no—
168C2×M16—61no—
169#124—21no—
170#124—21no—
171#124—21no—
172#124—21no—
173#117—61no—
174#117—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—31no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—64no—
193C4⋊C8—61no—
194C22⋊C8—21no—
195C22⋊C8—64no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—61no—
200#97#6712yescomplemented
201C4×C8—21no—
202C4×C8—61no—
203#95—31no—
204#95—31no—
205#95—61no—
206#95—61no—
207#95—61no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C2×C22⋊C411yescharacteristic
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—41no—
220C22×D3—41no—
221C3×Q8—81no—
222C3×Q8—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238#67—84no—
239C2×Dic3—22no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C4×D3—42no—
253C4×D3—42no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259C24—31no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—81no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—81no—
266C2×D4—122no—
267C2×D4—122no—
268C2×D4—241no—
269C22×C4—21no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4C2×C4×D311yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282M16—41no—
283M16—41no—
284M16—121no—
285M16—121no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—241no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—122no—
304C22⋊C4—122no—
305C22⋊C4—241no—
306C22⋊C4—241no—
307C42—61no—
308C42—61no—
309C42—122no—
310C42#23511yescharacteristic
311C42D3×D411yescharacteristic
312C42#23711yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3#38611yescharacteristic
324C22×C3#42111yescharacteristic
325C22×C3#42111yescharacteristic
326C2×D3—42no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—81no—
338Dic3—42no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—81no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—121no—
354C23—121no—
355C23D3×C22⋊C411yescharacteristic
356Q8—81no—
357Q8—81no—
358D4—81no—
359D4—81no—
360D4—244no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4#67⋊C411yescharacteristic
384C2×C4#88311yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4C4×#6711yescharacteristic
387C2×C4#93711yescharacteristic
388C2×C4#93811yescharacteristic
389C8—42no—
390C8—81no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—161no—
396C2×C3#188411yescharacteristic
397C2×C3#189711yescharacteristic
398C2×C3#191411yescharacteristic
399D3—81no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—122no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—241no—
414C22#482111yescharacteristic
415C22#482111yescharacteristic
416C22#495611yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—81no—
424C4—122no—
425C4—121no—
426C4—121no—
427C4—121no—
428C4—241no—
429C3#938911yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—161no—
433C2—241no—
434C2#7008911yescharacteristic
435C2#7038311yescharacteristic
436C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#386
prime radical#5223C4
cosocleC2×C3×C42C23
hypercenter#1527D3
hypocenterC3#9389
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #78518C2semidirect
3. C3×#1527C2semidirect
4. #71718C2semidirect
5. #70826C2semidirect
6. #67748C2semidirect
7. #67665C2semidirect
35. #1527D3semidirect
70. C3×#97D4semidirect
82. #803D4semidirect
200. #97#67semidirect
429. C3#9389semidirect

Maximal subgroups

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

subgroupindex
#830002
#785182
C3×#15272
#717182
#708262
#677482
#676652
#93893