Finite Group Databasebeta

← #10696215

Subgroups — #10696215

Order 768.

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

normal subgroups only

This group has 498 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×#3100C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic, complemented
3#82315C211yesmaximal, characteristic, complemented
4#71799C211yesmaximal, characteristic, complemented
5#71603C211yesmaximal, characteristic, complemented
6#70771C211yesmaximal, characteristic, complemented
7#67747C211yesmaximal, characteristic, complemented
8#10042—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#480—21no—
13C3×#474—21no—
14C3×#459—21no—
15C3×#447—21no—
16C3×C4×D8—22no—
17C3×#431C2211yescharacteristic
18#5218—21no—
19#5081—21no—
20#5071—21no—
21#5035—22no—
22#5019—21no—
23#5018—22no—
24#5008—21no—
25#5007—21no—
26#5007—21no—
27#4970—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4540—22no—
32#4525—22no—
33#4456—22no—
34#3235—31no—
35#3110—31no—
36#3100D311yescharacteristic
37#1536—31no—
38#1497—31no—
39#1466—31no—
40#1287—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—22no—
60#926—21no—
61#924—21no—
62#924—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#107—22no—
69C3×C4⋊C8—21no—
70C3×#102—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×C4×C8D412yes—
74#832—22no—
75#832—22no—
76#832—22no—
77#830—21no—
78#830—21no—
79#830D412yes—
80#810—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89#803D412yes—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#490—61no—
95#484—61no—
96#482—62no—
97#480—21no—
98#476—61no—
99#474—21no—
100#474—61no—
101#474—61no—
102#461—61no—
103#459—21no—
104#450—61no—
105#447—21no—
106#440—62no—
107C4×D8—22no—
108#432—31no—
109#432—31no—
110#432—31no—
111#431C2×D311yescharacteristic
112#329—62no—
113Q8⋊C8—62no—
114#323—62no—
115C24×C3—21no—
116C3×#41—82no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—82no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×Dic6—21no—
134C2×Dic6—21no—
135C3×D8—44no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149#217—41no—
150#214—44no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×#41—21no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×SD16—62no—
165C2×D8—22no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8#6711yescharacteristic
169#127#6711yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184#117—61no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C42C22×D311yescharacteristic
188#107—22no—
189#107—62no—
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#103—62no—
197#102—22no—
198#102—22no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—32no—
207#97—61no—
208#97—61no—
209C4×C8#6712yes—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—81no—
215C23×C3—81no—
216C23×C3#12111yescharacteristic
217C3×Q8—81no—
218C3×D4—22no—
219C3×D4—22no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—81no—
225C3×D4—82no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—82no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×Dic3—41no—
242C2×Dic3—41no—
243Dic6—42no—
244Dic6—42no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—21no—
250#41—82no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4D3×D411yescharacteristic
267C22×C4#23711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269SD16—124no—
270D8—44no—
271C2×C8—22no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—121no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—121no—
290C42—121no—
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42C2×#6711yescharacteristic
294C42C2×#6711yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3#39211yescharacteristic
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4#47212yes—
315Dic3—81no—
316Dic3—81no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—81no—
322C23—81no—
323C23#93911yescharacteristic
324Q8—81no—
325Q8—122no—
326Q8—122no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331D4—42no—
332D4—42no—
333D4—81no—
334D4—82no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4#93711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#94011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—42no—
353C8—122no—
354C8—122no—
355C8—241no—
356C2×C3—41no—
357C2×C3—82no—
358C2×C3—161no—
359C2×C3#200911yescharacteristic
360C2×C3#201011yescharacteristic
361C2×C3#201811yescharacteristic
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#522311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—81no—
381C4—241no—
382C4—241no—
383C4#516012yes—
384C3#1004211yescharacteristic, complemented, Sylow
385C2—41no—
386C2—82no—
387C2—161no—
388C2#7171511yescharacteristic
389C2#7204611yescharacteristic
390C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3#392
hypercenter#3100D3
hypocenterC3#10042
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #83274C2semidirect
3. #82315C2semidirect
4. #71799C2semidirect
5. #71603C2semidirect
6. #70771C2semidirect
7. #67747C2semidirect
384. C3#10042semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#832742
#823152
#717992
#716032
#707712
#677472
#100423