Finite Group Databasebeta

← #10691499

Subgroups — #10691499

Order 768.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#4996C211yesmaximal, characteristic
2#82210C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic
4#69972C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic
6#67687C211yesmaximal, characteristic
7#67617C211yesmaximal, characteristic
8#9240—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5207C2211yescharacteristic
14C2×#830C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5000—22no—
17#4999—21no—
18#4997—21no—
19#4996—21no—
20#4996—21no—
21#4996—21no—
22#4996—21no—
23C2×#810C411yescharacteristic, complemented
24C2×#810C411yescharacteristic, complemented
25C2×#808C411yescharacteristic, complemented
26C2×#807C411yescharacteristic, complemented
27#4926—21no—
28#4925—21no—
29C3×#339—21no—
30C3×#339—21no—
31#4469—21no—
32#4469—21no—
33#4452—22no—
34C2×#485—31no—
35#2924—31no—
36#1826D311yescharacteristic
37#1508—31no—
38#1472—31no—
39#1293—31no—
40#1282—31no—
41C23×C3×C4C2311yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—21no—
44C3×#117—21no—
45C3×#117—21no—
46C3×#117D411yescharacteristic
47C3×#117D411yescharacteristic
48C3×#117D411yescharacteristic
49C3×#117D411yescharacteristic
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×#217—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58#924—21no—
59#924—21no—
60#924—21no—
61#924—21no—
62C2×Dic3⋊C4—21no—
63C2×#208—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#830D411yescharacteristic
75#830D411yescharacteristic
76#830D412yes—
77#810—21no—
78#810—21no—
79#810—21no—
80#810—21no—
81#808—21no—
82#808—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—22no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88C4×C3⋊C8—22no—
89C2×C22⋊Q8—31no—
90C2×#117C2×D311yescharacteristic
91#485—61no—
92#485—61no—
93#485—61no—
94#485—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98C2×#107—31no—
99C2×#106—31no—
100#424—61no—
101C2×#103—31no—
102C2×#103—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8—31no—
107#405—61no—
108C4×C22⋊C4C2×D311yescharacteristic
109#339—21no—
110#339—21no—
111#328—62no—
112#324—61no—
113#324—61no—
114C24×C3#4111yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4C22⋊C411yescharacteristic
138C3×C4⋊C4C22⋊C411yescharacteristic
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—41no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—41no—
152C3×C42—41no—
153#217—41no—
154#217—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C22×Q8—31no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—61no—
174C22×C8—31no—
175C22×C8—31no—
176C22×C8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181#117—21no—
182#117—21no—
183#117D1211yescharacteristic
184#117D1211yescharacteristic
185#117#6711yescharacteristic
186#117#6711yescharacteristic
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4C4×D311yescharacteristic
190C2×C4⋊C4C4×D311yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C42—21no—
195C2×C42—21no—
196#107—61no—
197#107—61no—
198#106—61no—
199#106—61no—
200C4⋊C8—62no—
201C4⋊C8—62no—
202#103—61no—
203#103—61no—
204#103—61no—
205#103—61no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—31no—
209C22⋊C8—31no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#97—62no—
216C4×C8—62no—
217#95—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3C4×D411yescharacteristic
223C23×C3#12111yescharacteristic
224C23×C3#12311yescharacteristic
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—81no—
253Dic6—41no—
254Dic6—41no—
255Dic6—81no—
256C3×C8—81no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—42no—
260C3⋊C8—41no—
261C3⋊C8—41no—
262C24#23711yescharacteristic
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—121no—
266C2×Q8—121no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4C2×D1211yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23711yescharacteristic
279C22×C4C2×#6711yescharacteristic
280M16—41no—
281M16—41no—
282M16—121no—
283M16—121no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4(C2×D3)⋊C411yescharacteristic
301C4⋊C4(C2×D3)⋊C411yescharacteristic
302C4⋊C4(C2×D3)⋊C411yescharacteristic
303C4⋊C4(C2×D3)⋊C411yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—41no—
317C42—41no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#38611yescharacteristic
329C22×C3#44311yescharacteristic
330C22×C3#44411yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4—82no—
339C3×C4#41011yescharacteristic
340C3×C4#41011yescharacteristic
341Dic3—81no—
342Dic3—81no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23#67⋊C411yescharacteristic
348C23#88611yescharacteristic
349C23#93911yescharacteristic
350Q8—121no—
351Q8—121no—
352Q8—241no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—241no—
377C2×C4#67⋊C411yescharacteristic
378C2×C4#88211yescharacteristic
379C2×C4C2×(C2×D3)⋊C411yescharacteristic
380C2×C4#93911yescharacteristic
381C8—81no—
382C8—121no—
383C8—121no—
384C8—122no—
385C8—121no—
386C8—121no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3#187611yescharacteristic
392C2×C3#189911yescharacteristic
393C2×C3#190211yescharacteristic
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22#482311yescharacteristic
405C22#482411yescharacteristic
406C22#495811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—241no—
416C4—241no—
417C4#500611yescharacteristic
418C4#500611yescharacteristic
419C3#924011yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#6953511yescharacteristic
425C2#7037711yescharacteristic
426C2#7039911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9240
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
23. C2×#810C4semidirect
24. C2×#810C4semidirect
25. C2×#808C4semidirect
26. C2×#807C4semidirect
419. C3#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#49962
#822102
C3×#18262
#699722
#694452
#676872
#676172
#92403