Finite Group Databasebeta

← #10695961

Subgroups — #10695961

Order 768.

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

normal subgroups only

This group has 548 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#82388C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3#79310C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic, complemented
6#70770C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10002—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5221—21no—
15#5215—21no—
16#5210—22no—
17#5009—22no—
18#5008—21no—
19#5007—21no—
20#4972—21no—
21#4964—21no—
22#4949—21no—
23#4947—21no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4850—21no—
27#4849—21no—
28#4739—21no—
29#4722—21no—
30C4×Dic12—22no—
31C3×D4⋊C8—22no—
32#4467—22no—
33#4456—22no—
34#3104—31no—
35#3077—31no—
36#2580—31no—
37#1518—31no—
38#1487—31no—
39#1478D311yescharacteristic
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×Dic6D411yescharacteristic
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52C2×#217—22no—
53#926—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—22no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
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×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77C3×C4×C8D412yes—
78#832—22no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#818—22no—
84#816—22no—
85#816—22no—
86#808—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90#533—31no—
91C2×C4×Q8—31no—
92C2×C4×D4C2×D311yescharacteristic
93#484—61no—
94#478—62no—
95#477—61no—
96#476—61no—
97#474—61no—
98#462—61no—
99#451—61no—
100C4×Q16—62no—
101#431C2×D311yescharacteristic
102#431C2×D311yescharacteristic
103#428—61no—
104#423—31no—
105#418—61no—
106#399—61no—
107#393—61no—
108#390—61no—
109#389—61no—
110#385—62no—
111#382—31no—
112#328—62no—
113Q8⋊C8—62no—
114D4⋊C8—22no—
115C24×C3—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4Q1612yescomplemented
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—82no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—42no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—44no—
148#217—41no—
149#217—82no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—82no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166#208—41no—
167#208—41no—
168C2×C3⋊C8—22no—
169Dic12—44no—
170C22×Q8—31no—
171C22×D4D1211yescharacteristic
172C23×C4—21no—
173C2×Q16—62no—
174C2×M16—61no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177#125—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117—61no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4D1211yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#107—62no—
203#106—62no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—32no—
207C4⋊C8—61no—
208#103—62no—
209#103—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C4×C8—21no—
215C4×C8#6712yes—
216#95—32no—
217#95—62no—
218#95—61no—
219C23×C3—22no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—41no—
228C3×D4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—82no—
234C2×C3×C4—82no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4C2×Q1611yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4#13711yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256Dic6—22no—
257Dic6—22no—
258Dic6—41no—
259Dic6—41no—
260Dic6—42no—
261Dic6—81no—
262C3×C8—42no—
263C3×C8—81no—
264C3⋊C8—42no—
265C24—21no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—122no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4Dic1212yescomplemented
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—61no—
283C22×C4—82no—
284C22×C4C2×D1211yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287Q16—124no—
288M16—121no—
289M16—121no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—62no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—242no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—242no—
313C42—121no—
314C42—121no—
315C42—121no—
316C42#23511yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—82no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#39211yescharacteristic
328C22×C3#45111yescharacteristic
329C22×C3#45211yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—81no—
335C3×C4#45111yescharacteristic
336C3×C4#45211yescharacteristic
337C3×C4#46711yescharacteristic
338C3×C4#47011yescharacteristic
339Dic3—42no—
340Dic3—82no—
341Dic3—81no—
342Dic3—81no—
343Dic3—81no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23#88211yescharacteristic
350Q8—62no—
351Q8—62no—
352Q8—121no—
353Q8—121no—
354Q8—122no—
355Q8—241no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—62no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4#88211yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4#90411yescharacteristic
381C2×C4C2×Dic1211yescharacteristic
382C2×C4#90911yescharacteristic
383C2×C4#90911yescharacteristic
384C2×C4#93011yescharacteristic
385C2×C4#93711yescharacteristic
386C2×C4#94011yescharacteristic
387C8—42no—
388C8—81no—
389C8—122no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—82no—
393C2×C3#200111yescharacteristic
394C2×C3#200811yescharacteristic
395C2×C3#201011yescharacteristic
396C22—21no—
397C22—21no—
398C22—44no—
399C22—42no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22#473311yescharacteristic
404C22#473911yescharacteristic
405C22#496011yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—122no—
412C4—242no—
413C4—241no—
414C4—241no—
415C4—241no—
416C4#473311yescharacteristic
417C4#473911yescharacteristic
418C4#511611yescharacteristic
419C4#513711yescharacteristic
420C3#1000211yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—82no—
424C2#6996411yescharacteristic
425C2#7033111yescharacteristic
426C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1478D3
hypocenterC3#10002
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #79310C2semidirect
5. #70793C2semidirect
6. #70770C2semidirect
7. #67738C2semidirect
121. C2×C3×D4Q16semidirect
276. C2×D4Dic12semidirect
420. C3#10002semidirect

Maximal subgroups

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

subgroupindex
#823882
#822962
#793102
C3×#14782
#707932
#707702
#677382
#100023