Finite Group Databasebeta

← #10693040

Subgroups — #10693040

Order 768.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#79263C211yesmaximal, characteristic, complemented
2#74240C211yesmaximal, characteristic
3#70762C211yesmaximal, characteristic
4#70528C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9486—31nomaximal, Sylow
9C2×#897—21no—
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#382C2211yescharacteristic
15C2×#824—21no—
16#4987—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4830—22no—
20#4718—22no—
21C8×Dic6—22no—
22#4672C412yes—
23#4670—22no—
24#4661C412yes—
25#4554—21no—
26#4483—22no—
27#4459—22no—
28#2982—31no—
29#2536—31no—
30#1801—31no—
31#1486—31no—
32#1486—31no—
33#1286—31no—
34#1286D311yescharacteristic, complemented
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4C2×C412yes—
48C2×C4×Dic3D412yes—
49C2×#208—21no—
50#897—42no—
51#897—42no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8D412yes—
65C3×#95—21no—
66C3×#95C2×C412yes—
67C3⋊#95—22no—
68#830—21no—
69#824—42no—
70Dic3⋊C8—24no—
71C8×Dic3—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8D412yes—
76C2×C22⋊Q8—61no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79C8×Q8—62no—
80C8×Q8—62no—
81#431—31no—
82#431—62no—
83#431C2×D311yescharacteristic
84#423—31no—
85#423C2×D311yescharacteristic
86C2×#101—61no—
87#398—32no—
88#394—32no—
89#388—62no—
90#382C2×D311yescharacteristic
91#380—61no—
92#343—61no—
93D4⋊C8—62no—
94D4⋊C8—62no—
95C23×D3—21no—
96C22×C3×C4—22no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3C22⋊C412yes—
102C22×Dic3C22⋊C412yes—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12—42no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6D812yes—
117C2×Dic6SD1612yes—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128(C2×D3)⋊C4—44no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—44no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—44no—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139C4×Dic3—22no—
140C4×Dic3—22no—
141#208—41no—
142#208—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8—31no—
146C22×D4—31no—
147C23×C4—61no—
148C2×M16—21no—
149C2×M16—61no—
150C22⋊Q8—122no—
151C22⋊Q8—122no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C4×D4—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—62no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C42—32no—
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—64no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#101—122no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C4×C8—32no—
181C4×C8—64no—
182C4×C8D1212yes—
183#95—21no—
184#95—62no—
185#95C4×D312yes—
186C23×C3C2×C22⋊C411yescharacteristic
187C22×D3—22no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—41no—
193C22×D3—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4C2×D811yescharacteristic
206C2×C3×C4C2×SD1611yescharacteristic
207C2×C3×C4#13511yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3#10214yes—
217D12—22no—
218D12—22no—
219D12—41no—
220D12—41no—
221C4×D3—81no—
222C4×D3—81no—
223C4×D3—81no—
224C4×D3—81no—
225Dic6—22no—
226Dic6—22no—
227Dic6—22no—
228Dic6—22no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—61no—
234C2×Q8—32no—
235C2×Q8—32no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—122no—
243C22×C4—22no—
244C22×C4—32no—
245C22×C4—32no—
246C22×C4—61no—
247C22×C4—121no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4C2×C4×D311yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256M16—41no—
257M16—41no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—124no—
273C22⋊C4—124no—
274C22⋊C4—124no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—62no—
278C42—62no—
279C42C2×D1211yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23911yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3#38611yescharacteristic
286C22×C3C2×#10211yescharacteristic
287C22×C3#41611yescharacteristic
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—42no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—82no—
300C3×C4#46511yescharacteristic
301C3×C4#46511yescharacteristic
302C3×C4#46611yescharacteristic
303C3×C4#46611yescharacteristic
304Dic3—24no—
305Dic3—42no—
306Dic3—81no—
307C23—62no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23D3×C22⋊C411yescharacteristic
315Q8—62no—
316Q8—62no—
317Q8—62no—
318Q8—62no—
319D4—62no—
320D4—62no—
321D4—121no—
322D4—121no—
323C2×C4—34no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—122no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4C4×D1211yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#89311yescharacteristic
344C2×C4#89411yescharacteristic
345C2×C4#89511yescharacteristic
346C2×C4D3×D811yescharacteristic
347C2×C4D3×SD1611yescharacteristic
348C2×C4#91611yescharacteristic
349C2×C4#91911yescharacteristic
350C8—42no—
351C8—81no—
352C8—122no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#189111yescharacteristic
357C2×C3#189311yescharacteristic
358C2×C3#191211yescharacteristic
359D3—82no—
360C22—21no—
361C22—21no—
362C22—122no—
363C22—122no—
364C22—122no—
365C22—241no—
366C22—241no—
367C22—241no—
368C22—241no—
369C22#466511yescharacteristic
370C22D3×#10211yescharacteristic
371C22#480411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—64no—
377C4—82no—
378C4—122no—
379C4—241no—
380C4#477511yescharacteristic
381C4#480711yescharacteristic
382C4#486411yescharacteristic
383C4#488311yescharacteristic
384C3#948611yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—242no—
388C2#6870811yescharacteristic
389C2#6888311yescharacteristic
390C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9486
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
1. #79263C2semidirect
4. #70528C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
34. #1286D3semidirect
384. C3#9486semidirect

Maximal subgroups

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

subgroupindex
#792632
#742402
#707622
#705282
#704962
C3×#12862
#676312
#94863