Finite Group Databasebeta

← #10691020

Subgroups — #10691020

Order 768.

458 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 95 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#83316C211yesmaximal, characteristic
2#79265C211yesmaximal, characteristic, complemented
3C3×#1502C211yesmaximal, characteristic
4#71798C211yesmaximal, characteristic
5#69475C211yesmaximal, characteristic, complemented
6#67742C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9164—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5091C411yescharacteristic, complemented
14#5091C411yescharacteristic, complemented
15#5075C411yescharacteristic, complemented
16#5075C411yescharacteristic, complemented
17#5040—21no—
18#5033—21no—
19#5018—21no—
20#5018—21no—
21C2×#810—21no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4828—22no—
26#4719—21no—
27#4718—21no—
28#4679—21no—
29C8×Dic6—21no—
30C3×#327—21no—
31C3×#327—21no—
32#4540—22no—
33#4529—21no—
34#3112—31no—
35#2984—31no—
36#1763—31no—
37#1530—31no—
38#1502D311yescharacteristic
39#1302—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#127C2×C411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—21no—
54C2×#214—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C22×C3⋊C8—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8D411yescharacteristic
67C3×C22⋊C8D411yescharacteristic
68C3×C4×C8—21no—
69C3×#95—21no—
70#832—21no—
71#832—21no—
72#830—21no—
73#830—21no—
74#815—22no—
75Dic3⋊C8—22no—
76Dic3⋊C8—22no—
77C8×Dic3—22no—
78#810—24no—
79#810—21no—
80#810—21no—
81#807—21no—
82#807—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86#803—21no—
87#803—21no—
88#803—21no—
89#803D412yes—
90#532C2×D311yescharacteristic
91C2×C4×Q8—31no—
92#480—61no—
93#477—31no—
94#477—31no—
95#460—31no—
96#460—31no—
97#448—61no—
98#446—61no—
99#446—62no—
100C8×Q8—61no—
101#440—61no—
102#440—61no—
103#432—31no—
104#432—31no—
105#432—61no—
106#431—61no—
107#431C2×D311yescharacteristic
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4C2×D311yescharacteristic
111#340—61no—
112#329—62no—
113#327—21no—
114#327—21no—
115C3×#41—81no—
116C3×#41—81no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×Dic6—22no—
130C2×Dic6—21no—
131C2×Dic6—41no—
132C2×Dic6C22⋊C411yescharacteristic
133C2×Dic6C22⋊C411yescharacteristic
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42—81no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#214—42no—
148#214—42no—
149C4⋊Dic3—21no—
150C4⋊Dic3C22⋊C411yescharacteristic
151C4⋊Dic3C22⋊C411yescharacteristic
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C22×Q8—31no—
168C2×SD16—61no—
169C2×SD16—61no—
170C22×C8—61no—
171C4⋊Q8C4×D311yescharacteristic
172#127C4×D311yescharacteristic
173#124—21no—
174#121—21no—
175#121—21no—
176C4×Q8—31no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×Q8—31no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×D4—21no—
183C4×D4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—61no—
191#107—61no—
192C4⋊C8—21no—
193C4⋊C8—31no—
194C4⋊C8—31no—
195C4⋊C8—62no—
196C4⋊C8—62no—
197C4⋊C8—61no—
198#103—61no—
199#103—61no—
200#103—64no—
201#102—61no—
202#102—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8#6711yescharacteristic
206C22⋊C8#6711yescharacteristic
207#97—32no—
208#97—62no—
209#97—61no—
210#97—61no—
211#97—61no—
212C4×C8—21no—
213C4×C8—62no—
214#95—21no—
215C23×C3—41no—
216C23×C3#12111yescharacteristic
217C3×Q8—81no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—82no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4C2×C22⋊C411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12311yescharacteristic
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247Dic6—21no—
248Dic6—21no—
249Dic6—21no—
250Dic6—42no—
251Dic6#10412yes—
252C3×C8—41no—
253C3×C8—41no—
254C3⋊C8—42no—
255C3⋊C8—42no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258#41—81no—
259#41—81no—
260C2×Q8—21no—
261C2×Q8—31no—
262C2×Q8—31no—
263C2×Q8—62no—
264C2×Q8—61no—
265C2×Q8—121no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277SD16—122no—
278SD16—122no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—62no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—62no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—61no—
301C42—61no—
302C42—61no—
303C42—61no—
304C42—81no—
305C42C2×C4×D311yescharacteristic
306C42#23511yescharacteristic
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3#47111yescharacteristic
316C22×C3#47211yescharacteristic
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—82no—
323C3×C4—81no—
324C3×C4C2×#10411yescharacteristic
325C3×C4#42111yescharacteristic
326Dic3—22no—
327Dic3—41no—
328Dic3—41no—
329Dic3—41no—
330C23—41no—
331C23#93811yescharacteristic
332Q8—32no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—81no—
337Q8—122no—
338D4—41no—
339D4—41no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4D3×C22⋊C411yescharacteristic
362C2×C4#67⋊C411yescharacteristic
363C2×C4#88311yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4C4×#6711yescharacteristic
366C2×C4#93711yescharacteristic
367C8—41no—
368C8—41no—
369C8—122no—
370C8—122no—
371C8—121no—
372C8—121no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#187111yescharacteristic
377C2×C3#190511yescharacteristic
378C2×C3#190711yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#495611yescharacteristic
385C22#516011yescharacteristic
386C22#517711yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—62no—
393C4—82no—
394C4—81no—
395C4—121no—
396C4—121no—
397C4—121no—
398C4D3×#10411yescharacteristic
399C4#482111yescharacteristic
400C3#916411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2#7005211yescharacteristic
405C2#7021811yescharacteristic
406C2#7038111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9164
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #79265C2semidirect
5. #69475C2semidirect
6. #67742C2semidirect
7. #67621C2semidirect
13. #5091C4semidirect
14. #5091C4semidirect
15. #5075C4semidirect
16. #5075C4semidirect
400. C3#9164semidirect

Maximal subgroups

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

subgroupindex
#833162
#792652
C3×#15022
#717982
#694752
#677422
#676212
#91643