Finite Group Databasebeta

← #10697948

Subgroups — #10697948

Order 768.

516 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 87 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#83374C211yesmaximal, characteristic, complemented
2#83366C211yesmaximal, characteristic
3#79065C211yesmaximal, characteristic
4C3×#1994C211yesmaximal, characteristic, complemented
5#69396C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic
7C3⋊#1309C211yesmaximal, characteristic, complemented
8#10302—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C43C2211yescharacteristic
14#5134—22no—
15#5094—21no—
16#5090—21no—
17#5087—21no—
18#5083—21no—
19#5052—22no—
20#5049—22no—
21#5031—21no—
22#5027—21no—
23#5021—21no—
24#5020—21no—
25C2×#805—22no—
26#4924C2211yescharacteristic
27#4924C2211yescharacteristic
28#4850—21no—
29#4848—21no—
30#4846—21no—
31#4841—21no—
32C3×#339—22no—
33#4523—22no—
34#3255—31no—
35#3254—31no—
36#3254—31no—
37#1994—31no—
38#1994D311yescharacteristic, complemented
39#1962—31no—
40#1309—31no—
41C2×#235—21no—
42C2×C3×M16D412yes—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#216—22no—
56C2×#215—22no—
57#930—21no—
58#930—21no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208D412yes—
63C2×#208D412yes—
64C2×#204—22no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#870—21no—
70#869D411yescharacteristic
71#869D411yescharacteristic
72#869D411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77#835—22no—
78#835—22no—
79#821—22no—
80#818—22no—
81#816—22no—
82#810—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86#805—44no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91#533—31no—
92#533C2×D311yescharacteristic
93#532—31no—
94#477—61no—
95#477—61no—
96#475—61no—
97#475—61no—
98#474—61no—
99#474—61no—
100#470—62no—
101#468—62no—
102#468—62no—
103#462—61no—
104#462—61no—
105#461—61no—
106#461—61no—
107#460—61no—
108#460—61no—
109#423—31no—
110#423—31no—
111#423C2×D311yescharacteristic
112C2×#104—62no—
113C43C2×D311yescharacteristic
114#339—22no—
115#339—62no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123#235—42no—
124#235—42no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C3×M16—42no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—22no—
142C3×C42—22no—
143C3×C42—22no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—41no—
153#216—82no—
154#215—82no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166#208—42no—
167#208—42no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170#204—82no—
171C2×#41—61no—
172C2×Q16—62no—
173C2×SD16—62no—
174C2×SD16—62no—
175C2×M16—32no—
176C2×M16—32no—
177C2×M16#6712yes—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181C4⋊Q8D1211yescharacteristic
182C4⋊Q8D1211yescharacteristic
183#127—31no—
184#125—21no—
185#125—61no—
186#124—61no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191#121—61no—
192#121—61no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—61no—
198C4×D4—61no—
199#117—21no—
200#117—61no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C22×D311yescharacteristic
205#107—62no—
206#107—62no—
207#106—62no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214#104—124no—
215#103—62no—
216#103—62no—
217#103—62no—
218#103—62no—
219#102—62no—
220#102—62no—
221C23×C3#12711yescharacteristic
222C22×D3—41no—
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—41no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244#67—82no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248D12—42no—
249D12—41no—
250D12—41no—
251D12—41no—
252D12—41no—
253C4×D3—81no—
254C4×D3—81no—
255Dic6—42no—
256Dic6—41no—
257Dic6—41no—
258Dic6—41no—
259Dic6—41no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263#41—122no—
264#41—122no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—121no—
274C22×C4—41no—
275C22×C4—121no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—242no—
280SD16—242no—
281SD16—242no—
282M16—42no—
283M16—122no—
284M16—122no—
285C2×C8—22no—
286C2×C8—62no—
287C2×C8—62no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C22⋊C4—41no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—121no—
314C42C2×D1211yescharacteristic
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3#39211yescharacteristic
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—81no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—44no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4#45311yescharacteristic
332C3×C4#45411yescharacteristic
333C3×C4#45511yescharacteristic
334C3×C4#45511yescharacteristic
335C3×C4#45611yescharacteristic
336C3×C4#45611yescharacteristic
337Dic3—81no—
338Dic3—81no—
339Dic3—81no—
340C23—121no—
341C23#94011yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—41no—
345Q8—41no—
346Q8—122no—
347Q8—121no—
348Q8—121no—
349Q8—121no—
350Q8—121no—
351D4—122no—
352D4—121no—
353D4—121no—
354D4—121no—
355D4—121no—
356D4—242no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4#88211yescharacteristic
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#88411yescharacteristic
377C2×C4#93011yescharacteristic
378C2×C4#93711yescharacteristic
379C8—42no—
380C8—122no—
381C8—122no—
382C2×C3—41no—
383C2×C3#200611yescharacteristic
384C2×C3#200711yescharacteristic
385C2×C3#200711yescharacteristic
386D3—161no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—241no—
391C22—241no—
392C22—241no—
393C22#496011yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—81no—
399C4—81no—
400C4—241no—
401C4—241no—
402C4—241no—
403C4#482211yescharacteristic
404C4#482511yescharacteristic
405C4#482611yescharacteristic
406C4#482711yescharacteristic
407C4#506111yescharacteristic
408C4#509511yescharacteristic
409C3#1030211yescharacteristic, complemented, Sylow
410C2—41no—
411C2—481no—
412C2#6969411yescharacteristic
413C2#7039211yescharacteristic
414C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1994D3
hypocenterC3#10302
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
1. #83374C2semidirect
4. C3×#1994C2semidirect
5. #69396C2semidirect
7. C3⋊#1309C2semidirect
38. #1994D3semidirect
409. C3#10302semidirect

Maximal subgroups

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

subgroupindex
#833742
#833662
#790652
C3×#19942
#693962
#693952
C3⋊#13092
#103023