Finite Group Databasebeta

← #10696074

Subgroups — #10696074

Order 768.

598 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3112C211yesmaximal, characteristic, complemented
2#83289C211yesmaximal, characteristic
3#82503C211yesmaximal, characteristic, complemented
4#71813C211yesmaximal, characteristic, complemented
5#71623C211yesmaximal, characteristic, complemented
6#70808C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10020—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#951—21no—
12C2×#949—21no—
13C2×#875C2211yescharacteristic
14C3×#480—21no—
15C3×#477—21no—
16C3×#460—21no—
17C3×#448—21no—
18C3×#440—22no—
19C3×#432C2211yescharacteristic
20#5231—21no—
21#5086—21no—
22#5074—21no—
23#5059—22no—
24#4993—21no—
25#4991—21no—
26#4979—21no—
27#4940—22no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4537—22no—
32#4535—22no—
33#4464—22no—
34#3153—31no—
35#3112D311yescharacteristic, complemented
36#2632—31no—
37#1533—31no—
38#1514—31no—
39#1491—31no—
40#1288—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×SD16—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61#951—44no—
62#949—44no—
63C2×#216—22no—
64C2×#215—22no—
65C2×#214—22no—
66C2×#213—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×#208—21no—
73#875—21no—
74#875—21no—
75#875—21no—
76#875—21no—
77C3×#107—22no—
78C3×C4⋊C8—21no—
79C3×#103—22no—
80C3×#102—22no—
81C3×C22⋊C8—21no—
82C3×#97D412yes—
83C3⋊#95—22no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#803—21no—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93C2×AGL(1,ℤ/8)—61no—
94#532C2×D311yescharacteristic
95C2×#124—31no—
96C2×C4×Q8C2×D311yescharacteristic
97#497—61no—
98#496—61no—
99#495—62no—
100#480—21no—
101#477—21no—
102#460—21no—
103#448—21no—
104#440—22no—
105#432—31no—
106#432C2×D311yescharacteristic
107C4×M16—31no—
108#399—61no—
109#395—61no—
110#394—61no—
111#393—61no—
112#388—62no—
113#383—31no—
114#329—62no—
115#327—62no—
116Q8⋊C8—62no—
117C23×D3—21no—
118C3×#41—42no—
119C3×#41—42no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—42no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×SD16—44no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#216—44no—
159#215—44no—
160#214—44no—
161#213—44no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—82no—
168Dic3⋊C4—82no—
169#208—24no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×#41—21no—
173C22×Q8—31no—
174C22×Q8#6711yescharacteristic
175C22×D4—31no—
176#137—124no—
177AGL(1,ℤ/8)—124no—
178C2×Q16—62no—
179C2×SD16—22no—
180C2×SD16—62no—
181C2×SD16—62no—
182C2×D8—62no—
183C2×M16—61no—
184C4⋊Q8#6711yescharacteristic
185#127#6711yescharacteristic
186#124—21no—
187#124—61no—
188#124—61no—
189#124—61no—
190#124—61no—
191#121—21no—
192#121—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—21no—
198C4×D4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—62no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—62no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208#107—22no—
209C4⋊C8—21no—
210C4⋊C8—61no—
211#103—22no—
212#102—22no—
213C22⋊C8—21no—
214C22⋊C8—61no—
215#97—61no—
216#97—61no—
217#97#6712yes—
218C4×C8—32no—
219#95—32no—
220#95—62no—
221#95—61no—
222#95—61no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—41no—
228C22×D3—42no—
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—42no—
238C3×D4—81no—
239C3×D4—82no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261D12—42no—
262D12—81no—
263Dic6—42no—
264Dic6—81no—
265C3×C8—42no—
266C3⋊C8—24no—
267C3⋊C8—81no—
268C24—61no—
269#41—42no—
270#41—42no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—122no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—122no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—62no—
289C22×C4—61no—
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4C2×#6711yescharacteristic
293Q16—124no—
294SD16—44no—
295SD16—124no—
296SD16—124no—
297D8—124no—
298M16—64no—
299C2×C8—22no—
300C2×C8—62no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—242no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—124no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—242no—
317C42—41no—
318C42—41no—
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39211yescharacteristic
329C22×C3#45312yes—
330C2×D3—84no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—42no—
341C3×C4—81no—
342C3×C4—81no—
343Dic3—82no—
344Dic3—81no—
345C23—41no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—121no—
350C23#93711yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8—41no—
356Q8—41no—
357Q8—122no—
358Q8—241no—
359D4—41no—
360D4—42no—
361D4—41no—
362D4—81no—
363D4—82no—
364D4—122no—
365D4—241no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4#93711yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4#94011yescharacteristic
386C2×C4#95311yescharacteristic
387C8—42no—
388C8—64no—
389C8—241no—
390C2×C3—22no—
391C2×C3—161no—
392C2×C3#200211yescharacteristic
393C2×C3#200911yescharacteristic
394C2×C3#201011yescharacteristic
395D3—161no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—244no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22#519812yes—
406C22#522311yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—81no—
415C4—81no—
416C4—242no—
417C4—241no—
418C3#1002011yescharacteristic, complemented, Sylow
419C2—22no—
420C2—161no—
421C2—481no—
422C2#7171511yescharacteristic
423C2#7209411yescharacteristic
424C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3112C2
socleC22×C3#392
prime radical#83289C2
hypercenter#3112D3
hypocenterC3#10020
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. C3×#3112C2semidirect
3. #82503C2semidirect
4. #71813C2semidirect
5. #71623C2semidirect
6. #70808C2semidirect
7. #67752C2semidirect
35. #3112D3semidirect
418. C3#10020semidirect

Maximal subgroups

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

subgroupindex
C3×#31122
#832892
#825032
#718132
#716232
#708082
#677522
#100203