Finite Group Databasebeta

← #10707992

Subgroups — #10707992

Order 768.

456 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 57 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#78128C211yesmaximal, characteristic
2C3×#1614C211yesmaximal, characteristic, complemented
3#71897C211yesmaximal, characteristic
4#71891C211yesmaximal, characteristic, complemented
5#67651C211yesmaximal, characteristic, complemented
6#67451C211yesmaximal, characteristic, complemented
7#67444C211yesmaximal, characteristic
8#12218—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C3×#484—21no—
11C3×#399C2211yescharacteristic
12#5044—21no—
13#4994—21no—
14#4994—21no—
15#4994—21no—
16#4955—21no—
17#4955—21no—
18#4955C2211yescharacteristic
19#4949—21no—
20#4949C2211yescharacteristic
21#4676—21no—
22C3×#328—21no—
23C3×#323C2211yescharacteristic
24#4538—21no—
25#4526—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2842—31no—
29#2029—31no—
30#1614—31no—
31#1614D311yescharacteristic, complemented
32#1352—31no—
33#1352—31no—
34#1316—31no—
35C24×D3D411yescharacteristic
36C3×C22⋊Q8—21no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—41no—
39C3×C22⋊Q8—41no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217D411yescharacteristic
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—41no—
55C2×(C2×D3)⋊C4—41no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C3×#107—41no—
62C3×#103—41no—
63C3×C22⋊C8—21no—
64C3×#95D411yescharacteristic
65#835—41no—
66C3⋊#95—21no—
67C3⋊#95D411yescharacteristic
68#830—21no—
69#807—41no—
70#543C2×D311yescharacteristic
71#484—21no—
72#484—61no—
73#399—31no—
74#399—61no—
75#399C2×D311yescharacteristic
76#396—61no—
77#396—61no—
78#396—61no—
79#396—61no—
80#379—31no—
81#379—61no—
82#379—61no—
83#342—31no—
84#342—61no—
85#328—21no—
86#328—61no—
87#323—31no—
88#323C2×D311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C23×D3—21no—
91C23×D3—21no—
92C23×D3—41no—
93C23×D3—41no—
94C23×D3—81no—
95C23×D3—81no—
96C23×D3Q1612yes—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—81no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—81no—
120#217—41no—
121#217—41no—
122#217—41no—
123#217—81no—
124(C2×D3)⋊C4—81no—
125(C2×D3)⋊C4—81no—
126(C2×D3)⋊C4—81no—
127(C2×D3)⋊C4—81no—
128(C2×D3)⋊C4—81no—
129(C2×D3)⋊C4—81no—
130(C2×D3)⋊C4—82no—
131(C2×D3)⋊C4—82no—
132C4⋊Dic3—41no—
133C4⋊Dic3—81no—
134Dic3⋊C4—81no—
135Dic3⋊C4—81no—
136Dic3⋊C4—81no—
137C2×C3⋊C8—41no—
138C25—31no—
139C22⋊Q8—21no—
140C22⋊Q8—21no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—121no—
162C2×C22⋊C4—121no—
163C2×C22⋊C4C22×D311yescharacteristic
164#107—41no—
165#107—121no—
166#103—41no—
167#103—121no—
168C22⋊C8—21no—
169C22⋊C8—61no—
170#95—31no—
171#95—61no—
172#95#6711yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#12011yescharacteristic
178C23×C3C2×Q1611yescharacteristic
179C23×C3#13711yescharacteristic
180C22×D3—22no—
181C22×D3—42no—
182C22×D3—41no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—82no—
191C22×D3—81no—
192C22×D3—82no—
193C22×D3—84no—
194C22×D3—81no—
195C22×D3—81no—
196C3×Q8—81no—
197C3×Q8—81no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C3×C8—81no—
225C3⋊C8—81no—
226C24—32no—
227C24—61no—
228C24—61no—
229C24—121no—
230C24—121no—
231C24—241no—
232C24—241no—
233C24D3×D411yescharacteristic
234C2×Q8—21no—
235C2×Q8—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—121no—
244C22×C4D3×D411yescharacteristic
245C22×C4C2×#6711yescharacteristic
246C2×C8—41no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—121no—
255C4⋊C4—241no—
256C4⋊C4—241no—
257C4⋊C4—241no—
258C4⋊C4—241no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—241no—
267C22⋊C4—241no—
268C22⋊C4—241no—
269C22⋊C4—241no—
270C22⋊C4—241no—
271C22⋊C4—241no—
272C22⋊C4—242no—
273C22⋊C4—242no—
274C22⋊C4—241no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—81no—
284C22×C3#45111yescharacteristic
285C22×C3#45511yescharacteristic
286C22×C3B(4,𝔽2)11yescharacteristic
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—44no—
290C2×D3—42no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—82no—
294C2×D3—82no—
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—82no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307Dic3—161no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—62no—
313C23—122no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—122no—
319C23—122no—
320C23—121no—
321C23—121no—
322C23—241no—
323C23—242no—
324C23—242no—
325C23—241no—
326C23—241no—
327C23—244no—
328C23D3×Q1611yescharacteristic
329C23#91811yescharacteristic
330C23#93711yescharacteristic
331Q8—81no—
332Q8—81no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—241no—
359C8—81no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3#160211yescharacteristic
366C2×C3#220011yescharacteristic
367C2×C3#220011yescharacteristic
368D3—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—81no—
378C22—122no—
379C22—122no—
380C22—124no—
381C22—122no—
382C22—241no—
383C22—241no—
384C22—242no—
385C22—242no—
386C22—242no—
387C22—241no—
388C22—241no—
389C22#503211yescharacteristic
390C22#518611yescharacteristic
391C22#520211yescharacteristic
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—241no—
399C4—241no—
400C4—241no—
401C4—481no—
402C3#1221811yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—41no—
406C2—41no—
407C2—242no—
408C2#7098011yescharacteristic
409C2#7098511yescharacteristic
410C2#7100111yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1614C2
socleC22×C3B(4,𝔽2)
prime radicalC24×D3D4
hypercenter#1614D3
hypocenterC3#12218
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1614

Complements

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

NH ≅ G/Ntype
2. C3×#1614C2semidirect
4. #71891C2semidirect
5. #67651C2semidirect
6. #67451C2semidirect
31. #1614D3semidirect
402. C3#12218semidirect

Maximal subgroups

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

subgroupindex
#781282
C3×#16142
#718972
#718912
#676512
#674512
#674442
#122183