Finite Group Databasebeta

← #10697297

Subgroups — #10697297

Order 768.

508 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3141C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#83222C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71779C211yesmaximal, characteristic, complemented
6#69868C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#489—22no—
13C3×#487—21no—
14C3×#486—21no—
15C3×#417—22no—
16C3×C4×M16C2211yescharacteristic
17#5228—22no—
18#5094—21no—
19#5092—21no—
20#5090—22no—
21#5079—21no—
22#5075—21no—
23#5041—22no—
24#5019—21no—
25#5008—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4555—22no—
30#4537—22no—
31#4529—22no—
32#3246—31no—
33#3209—31no—
34#3141D311yescharacteristic
35#1987—31no—
36#1986—31no—
37#1502—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#216—24no—
57C2×#214—22no—
58#926—21no—
59C2×C4⋊Dic3—21no—
60#924—21no—
61#924—21no—
62C2×C4×Dic3—21no—
63C22×C3⋊C8D412yes—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#103—24no—
70C3×#102—24no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#835—24no—
74#832—22no—
75C3⋊#95—21no—
76#830D412yes—
77#810—22no—
78#808—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#532C2×D311yescharacteristic
85C2×C4⋊Q8C2×D311yescharacteristic
86#494—61no—
87#489—22no—
88#487—21no—
89#487—61no—
90#486—21no—
91#476—61no—
92#475—61no—
93#465—62no—
94#462—62no—
95#461—61no—
96#460—61no—
97#451—62no—
98#431—31no—
99C2×C4⋊C8—31no—
100#417—22no—
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#327—62no—
106C3×#41—42no—
107C3×#41—42no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—41no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×M16—22no—
125C3×M16—22no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140#216—84no—
141#214—82no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—42no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C2×#41—21no—
156C22×Q8—21no—
157C2×Q16—64no—
158C2×SD16—62no—
159C2×M16#6712yes—
160C22×C8—32no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8#6712yes—
167C4⋊Q8#6711yescharacteristic
168#127#6711yescharacteristic
169#125—61no—
170#124—21no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—21no—
174#121—21no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×D4—21no—
178C4×D4—21no—
179#117—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#106—62no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—24no—
190#103—62no—
191#103—64no—
192#102—24no—
193#102—62no—
194C22⋊C8—32no—
195#97—21no—
196C4×C8—21no—
197C4×C8—61no—
198#95—61no—
199C23×C3—41no—
200C23×C3#12711yescharacteristic
201C3×Q8—44no—
202C3×Q8—42no—
203C3×Q8—82no—
204C3×D4—42no—
205C3×D4—41no—
206C3×D4—41no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—82no—
216C2×C3×C4—82no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#13512yes—
224C2×C3×C4#13712yes—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230Dic6—42no—
231Dic6—41no—
232Dic6—41no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236#41—42no—
237#41—42no—
238C2×Q8—22no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—82no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—121no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—244no—
255SD16—242no—
256M16—22no—
257M16—22no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C42—121no—
280C42—122no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39211yescharacteristic
291C22×C3#49511yescharacteristic
292C22×C3#49711yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—82no—
297C3×C4—81no—
298C3×C4#45212yes—
299C3×C4#45511yescharacteristic
300C3×C4#45611yescharacteristic
301C3×C4#46412yes—
302Dic3—82no—
303Dic3—81no—
304Dic3—81no—
305C23—41no—
306C23#94011yescharacteristic
307Q8—44no—
308Q8—42no—
309Q8—82no—
310Q8—122no—
311Q8—121no—
312Q8—121no—
313D4—41no—
314D4—41no—
315D4—42no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4#93711yescharacteristic
332C2×C4#93711yescharacteristic
333C2×C4#93811yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#93911yescharacteristic
336C2×C4#95012yes—
337C2×C4#95112yes—
338C2×C4#95311yescharacteristic
339C8—42no—
340C8—122no—
341C8—122no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—161no—
345C2×C3#200711yescharacteristic
346C2×C3#200811yescharacteristic
347C2×C3#201511yescharacteristic
348C22—21no—
349C22—21no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22#505911yescharacteristic
354C22#506011yescharacteristic
355C22#522311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—82no—
361C4—242no—
362C4—241no—
363C4—241no—
364C4#505412yes—
365C4#520211yescharacteristic
366C4#520511yescharacteristic
367C4#523912yes—
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—161no—
372C2#7172911yescharacteristic
373C2#7209911yescharacteristic
374C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3141C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3141D3
hypocenterC3#10198
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
3. #83222C2semidirect
5. #71779C2semidirect
6. #69868C2semidirect
7. #67668C2semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#833202
#832222
#718112
#717792
#698682
#676682
#101983