Finite Group Databasebeta

← #10697687

Subgroups — #10697687

Order 768.

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

normal subgroups only

This group has 474 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
1#83373C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic
3#79068C211yesmaximal, characteristic
4C3×#1493C211yesmaximal, characteristic
5#69788C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic
8#10259—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5121—21no—
15#5120—21no—
16#5094—21no—
17#5091—21no—
18#5091—21no—
19#5090—21no—
20#5063—22no—
21#5051—22no—
22#5031—21no—
23#5021—21no—
24#5004—21no—
25#4995—21no—
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4851—21no—
29#4850—21no—
30#4829—22no—
31C3×Q8⋊C8—22no—
32#4550—22no—
33#4456—22no—
34#3249—31no—
35#3078—31no—
36#3078—31no—
37#1984—31no—
38#1493—31no—
39#1493D311yescharacteristic
40#1292—31no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—22no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61C22×C3⋊C8D412yes—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75#835—22no—
76#818—22no—
77#817—22no—
78#816—22no—
79#816—22no—
80#810—22no—
81#810—22no—
82#810—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×C4×Q8—31no—
93#479—62no—
94#479—62no—
95#477—61no—
96#477—61no—
97#477—61no—
98#477—61no—
99#475—61no—
100#475—61no—
101#467—62no—
102#462—61no—
103#462—61no—
104#428—61no—
105#428—61no—
106#423—31no—
107#423C2×D311yescharacteristic
108C2×C4⋊C8—31no—
109#419—61no—
110#419—61no—
111C2×C4×C8C2×D311yescharacteristic
112#336—62no—
113Q8⋊C8—22no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6Q1612yescomplemented
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#217—41no—
149#216—82no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C22×Q8—31no—
169C2×Q16—62no—
170C2×M16—21no—
171C2×M16—61no—
172C22×C8—21no—
173C22×C8—32no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8D1211yescharacteristic
177C4⋊Q8D1211yescharacteristic
178#125—21no—
179#125—61no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192#117—21no—
193#117—61no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C42—61no—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#107—62no—
200#107—62no—
201#106—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C4⋊C8#6712yes—
209#103—62no—
210#103—62no—
211#103—62no—
212#103—62no—
213#103—62no—
214#103—62no—
215C4×C8—21no—
216C4×C8—21no—
217C23×C3#12111yescharacteristic
218C3×Q8—42no—
219C3×Q8—41no—
220C3×Q8—41no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×Q1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243Dic6—22no—
244Dic6—22no—
245Dic6—41no—
246Dic6—41no—
247Dic6—42no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3×C8—42no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—32no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4#23511yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269Q16—242no—
270M16—41no—
271M16—41no—
272M16—121no—
273M16—121no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—122no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C42—41no—
300C42—121no—
301C42—121no—
302C42—121no—
303C42C2×D1211yescharacteristic
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3#39211yescharacteristic
310C22×C3#46711yescharacteristic
311C22×C3#47011yescharacteristic
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4#45111yescharacteristic
318C3×C4#45111yescharacteristic
319C3×C4#45211yescharacteristic
320C3×C4#45211yescharacteristic
321C3×C4#45511yescharacteristic
322C3×C4#45611yescharacteristic
323Dic3—42no—
324Dic3—81no—
325Dic3—81no—
326Dic3—81no—
327C23#88411yescharacteristic
328Q8—42no—
329Q8—41no—
330Q8—41no—
331Q8—62no—
332Q8—62no—
333Q8—122no—
334Q8—121no—
335Q8—121no—
336Q8—121no—
337Q8—121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—62no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4#88211yescharacteristic
351C2×C4#88211yescharacteristic
352C2×C4#88411yescharacteristic
353C2×C4#91511yescharacteristic
354C2×C4#91611yescharacteristic
355C2×C4D3×Q1611yescharacteristic
356C2×C4#91811yescharacteristic
357C2×C4#93011yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#94011yescharacteristic
360C8—42no—
361C8—42no—
362C8—122no—
363C8—122no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#200111yescharacteristic
367C2×C3#200711yescharacteristic
368C2×C3#201711yescharacteristic
369C22—21no—
370C22—21no—
371C22#481311yescharacteristic
372C22#481611yescharacteristic
373C22#496011yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—81no—
379C4—122no—
380C4—241no—
381C4—241no—
382C4—241no—
383C4#480811yescharacteristic
384C4#480911yescharacteristic
385C4#482511yescharacteristic
386C4#482611yescharacteristic
387C4#517311yescharacteristic
388C4#518611yescharacteristic
389C3#1025911yescharacteristic, complemented, Sylow
390C2—21no—
391C2—21no—
392C2#7001311yescharacteristic
393C2#7027311yescharacteristic
394C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1493D3
hypocenterC3#10259
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
129. C2×Dic6Q16semidirect
389. C3#10259semidirect

Maximal subgroups

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

subgroupindex
#833732
#833672
#790682
C3×#14932
#697882
#697382
#677112
#102593