Finite Group Databasebeta

← #10696058

Subgroups — #10696058

Order 768.

528 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 99 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#82467C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic
3#79310C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic
5#70515C211yesmaximal, characteristic
6#70500C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic
8#10017—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5230—21no—
15#5228—21no—
16#5031—21no—
17C4×#216—22no—
18#5021—21no—
19#4983—21no—
20#4981—21no—
21#4979—21no—
22#4970—21no—
23#4932C2211yescharacteristic
24C4×Dic3⋊C4—22no—
25#4925C2211yescharacteristic
26#4850—21no—
27#4849—21no—
28#4739—21no—
29#4722—21no—
30C4×Dic12—22no—
31C3×Q8⋊C8—22no—
32#4555—22no—
33#4484—22no—
34#3104—31no—
35#3104—31no—
36#2399—31no—
37#1995—31no—
38#1487—31no—
39#1487D311yescharacteristic
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53#926—21no—
54C2×C4⋊Dic3D411yescharacteristic
55#924—21no—
56#924—21no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D412yes—
62C2×Dic12—22no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#818—22no—
84#816—22no—
85#816—22no—
86#810—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8—31no—
92C2×C4×Q8C2×D311yescharacteristic
93#484—61no—
94#484—61no—
95#477—61no—
96#477—61no—
97#462—61no—
98#462—61no—
99#451—61no—
100#451—61no—
101C4×Q16—62no—
102C4×Q16—62no—
103#431—31no—
104#431C2×D311yescharacteristic
105#423C2×D311yescharacteristic
106#398—61no—
107#398—61no—
108#395—61no—
109#395—61no—
110C4×C4⋊C4—62no—
111#376—31no—
112#339—62no—
113Q8⋊C8—22no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—42no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—42no—
131C3×M16—42no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#217—41no—
145#216—44no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—44no—
153Dic3⋊C4—44no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—22no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3Q1614yescomplemented
163C2×C3⋊C8—22no—
164Dic12—44no—
165C22×Q8—31no—
166C22×Q8D1211yescharacteristic
167C2×Q16—62no—
168C2×Q16—62no—
169C2×M16—21no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#125—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185#117—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C42—21no—
193C2×C42—32no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#107—62no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202#103—62no—
203#103—62no—
204#103—62no—
205#103—62no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208C4×C8—32no—
209C4×C8#6712yes—
210#95—62no—
211#95—62no—
212#95—61no—
213#95—61no—
214C23×C3#12011yescharacteristic
215C3×Q8—22no—
216C3×Q8—22no—
217C3×Q8—41no—
218C3×Q8—41no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×Q1612yes—
233C2×C3×C4#13712yes—
234C2×Dic3—22no—
235C2×Dic3—24no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244Dic6—22no—
245Dic6—22no—
246Dic6—41no—
247Dic6—41no—
248Dic6—42no—
249Dic6—42no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—42no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—122no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—62no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4C2×D1211yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269Q16—124no—
270Q16—124no—
271M16—42no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—62no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—124no—
286C4⋊C4—124no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C22⋊C4—121no—
292C42—34no—
293C42—41no—
294C42—41no—
295C42—62no—
296C42—121no—
297C42—121no—
298C42—121no—
299C42#23511yescharacteristic
300C42D3×D411yescharacteristic
301C42D3×D411yescharacteristic
302C42C2×#6711yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#39211yescharacteristic
306C22×C3#45112yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—81no—
313C3×C4#46212yes—
314C3×C4#49411yescharacteristic
315C3×C4#49711yescharacteristic
316Dic3—44no—
317Dic3—42no—
318Dic3—81no—
319Dic3—81no—
320Dic3—81no—
321C23#88211yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—62no—
327Q8—62no—
328Q8—121no—
329Q8—121no—
330Q8—122no—
331Q8—122no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—62no—
340C2×C4—64no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4#88211yescharacteristic
350C2×C4#88411yescharacteristic
351C2×C4#88411yescharacteristic
352C2×C4#91512yes—
353C2×C4D3×Q1612yes—
354C2×C4#93011yescharacteristic
355C2×C4#93711yescharacteristic
356C2×C4#94011yescharacteristic
357C8—42no—
358C8—81no—
359C8—122no—
360C2×C3—22no—
361C2×C3#201011yescharacteristic
362C2×C3#201011yescharacteristic
363C2×C3#201611yescharacteristic
364C22—21no—
365C22—21no—
366C22#480912yes—
367C22#496011yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—124no—
375C4—122no—
376C4—241no—
377C4—241no—
378C4—241no—
379C4#479512yes—
380C4#516711yescharacteristic
381C4#518311yescharacteristic
382C3#1001711yescharacteristic, complemented, Sylow
383C2—22no—
384C2#6999311yescharacteristic
385C2#7028311yescharacteristic
386C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1487D3
hypocenterC3#10017
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
162. C4×Dic3Q16semidirect
382. C3#10017semidirect

Maximal subgroups

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

subgroupindex
#824672
#824472
#793102
C3×#14872
#705152
#705002
#677372
#100173