Finite Group Databasebeta

← #10697410

Subgroups — #10697410

Order 768.

468 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3235C211yesmaximal, characteristic
2#83374C211yesmaximal, characteristic, complemented
3#83274C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71816C211yesmaximal, characteristic
6#70836C211yesmaximal, characteristic, complemented
7#67758C211yesmaximal, characteristic, complemented
8#10216—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—21no—
13C3×#490—21no—
14C3×#482—22no—
15C3×#476—21no—
16C3×#474—21no—
17C3×#432C2211yescharacteristic
18C2×#837—22no—
19#5094—21no—
20#5090—21no—
21#5081—21no—
22#5071—21no—
23#5052—22no—
24#5035—22no—
25#5031—21no—
26#5020—21no—
27#5008—21no—
28#5007—21no—
29#4932C2211yescharacteristic
30#4926C2211yescharacteristic
31#4924C2211yescharacteristic
32#4538—22no—
33#4467—22no—
34#3255—31no—
35#3235—31no—
36#3235D311yescharacteristic
37#1963—31no—
38#1506—31no—
39#1506—31no—
40#1307—31no—
41C2×C3×#41—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—22no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×C4×Dic3—21no—
63C2×#208D412yes—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#107—22no—
70C3×#106—22no—
71C3×C4⋊C8—21no—
72C3×#102—22no—
73C3×#102—22no—
74C3×C22⋊C8D412yes—
75C3×#97—21no—
76#837—44no—
77#835—22no—
78#832—22no—
79#832—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830D412yes—
83#810—22no—
84#808—22no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91#533—31no—
92#533C2×D311yescharacteristic
93#532C2×D311yescharacteristic
94#501—21no—
95#501—61no—
96#490—21no—
97#490—61no—
98#482—22no—
99#482—62no—
100#476—21no—
101#476—61no—
102#475—61no—
103#475—61no—
104#474—21no—
105#474—61no—
106#470—62no—
107#462—61no—
108#462—61no—
109#432—31no—
110#432C2×D311yescharacteristic
111#423—31no—
112C2×#104—62no—
113#376—31no—
114#328—62no—
115#328—62no—
116C3×#41—42no—
117C3×#41—42no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—41no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—41no—
150#216—82no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—22no—
158C4×Dic3—41no—
159C4×Dic3—41no—
160#208—42no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×#41—21no—
164C2×Q16—62no—
165C2×M16—32no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8#6711yescharacteristic
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#127#6711yescharacteristic
172#125—21no—
173#125—61no—
174#124—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179#121—21no—
180#121—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—21no—
186C4×D4—21no—
187#117—21no—
188#117—61no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—22no—
192#107—62no—
193#106—22no—
194#106—62no—
195#106—62no—
196C4⋊C8—21no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#104—124no—
201#103—62no—
202#103—62no—
203#102—22no—
204#102—22no—
205#102—62no—
206#102—62no—
207C22⋊C8—32no—
208C22⋊C8#6712yes—
209#97—21no—
210#97—61no—
211#95—61no—
212#95—61no—
213C23×C3—41no—
214C23×C3#12711yescharacteristic
215C3×Q8—42no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×Q8—81no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—82no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243Dic6—41no—
244Dic6—41no—
245Dic6—81no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249#41—42no—
250#41—42no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—121no—
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—242no—
266M16—122no—
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—121no—
288C42—41no—
289C42—62no—
290C42—121no—
291C42—121no—
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C42C2×#6711yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3#39211yescharacteristic
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#45311yescharacteristic
312C3×C4#45411yescharacteristic
313Dic3—82no—
314Dic3—81no—
315Dic3—81no—
316C23—41no—
317C23#94011yescharacteristic
318Q8—42no—
319Q8—41no—
320Q8—41no—
321Q8—81no—
322Q8—121no—
323Q8—121no—
324Q8—241no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4#93711yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#93811yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#93911yescharacteristic
348C2×C4#95311yescharacteristic
349C8—42no—
350C8—122no—
351C8—122no—
352C2×C3—41no—
353C2×C3—161no—
354C2×C3#200611yescharacteristic
355C2×C3#200911yescharacteristic
356C2×C3#200911yescharacteristic
357C22—21no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#522311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—242no—
373C4—241no—
374C4—241no—
375C4#519811yescharacteristic
376C4#520111yescharacteristic
377C3#1021611yescharacteristic, complemented, Sylow
378C2—41no—
379C2—161no—
380C2#7186911yescharacteristic
381C2#7209811yescharacteristic
382C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3235C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3235D3
hypocenterC3#10216
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3235

Complements

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

NH ≅ G/Ntype
2. #83374C2semidirect
4. #71836C2semidirect
6. #70836C2semidirect
7. #67758C2semidirect
377. C3#10216semidirect

Maximal subgroups

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

subgroupindex
C3×#32352
#833742
#832742
#718362
#718162
#708362
#677582
#102163