Finite Group Databasebeta

← #10695937

Subgroups — #10695937

Order 768.

554 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 97 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#83315C211yesmaximal, characteristic
2#83314C211yesmaximal, characteristic
3#74191C211yesmaximal, characteristic, complemented
4C3×#1517C211yesmaximal, characteristic
5#70770C211yesmaximal, characteristic, complemented
6#70766C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9999—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×C4×Dic6C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5232—21no—
16#5094—21no—
17#5090—21no—
18#5075—21no—
19#5075—21no—
20#5054—22no—
21#5040—21no—
22#5034—21no—
23C4×#214—22no—
24#4984—21no—
25#4973—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4659—21no—
29#4656—21no—
30#4653—22no—
31C3×#327—22no—
32#4484—22no—
33#4456—22no—
34#3103—31no—
35#3037—31no—
36#2581—31no—
37#1517D311yescharacteristic
38#1487—31no—
39#1479—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×Dic6D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53#951—42no—
54#951—42no—
55C2×#216—22no—
56C2×#214—22no—
57C2×#214—22no—
58C2×C4⋊Dic3—22no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—21no—
65C2×#208—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77C3×#95—21no—
78C3×#95D412yes—
79#835—22no—
80#832—22no—
81#832—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85#830—21no—
86#810—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yes—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93#532C2×D311yescharacteristic
94C2×C4×Q8—31no—
95C2×C4×Q8—31no—
96#481—61no—
97#475—61no—
98#464—62no—
99#462—61no—
100#460—61no—
101#460—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423—31no—
107#417—61no—
108#400—61no—
109#395—61no—
110#391—61no—
111#389—61no—
112#387—62no—
113#382C2×D311yescharacteristic
114#327—22no—
115Q8⋊C8—62no—
116Q8⋊C8—62no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—22no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—42no—
143C2×Dic6SD1612yescomplemented
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#216—44no—
156#214—44no—
157#214—44no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—44no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×#41—21no—
176C22×Q8—31no—
177C22×Q8—31no—
178#137—122no—
179#137—122no—
180C2×Q16—62no—
181C2×SD16—62no—
182C2×SD16—62no—
183C2×M16—61no—
184C4⋊Q8D1211yescharacteristic
185#127D1211yescharacteristic
186#124—21no—
187#121—21no—
188#121—21no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—21no—
198C4×D4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#106—62no—
209C4⋊C8—21no—
210C4⋊C8—32no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213#103—62no—
214#103—62no—
215#102—62no—
216#102—62no—
217C22⋊C8—21no—
218C22⋊C8—61no—
219C4×C8—21no—
220C4×C8—32no—
221#95—21no—
222#95—62no—
223#95—62no—
224#95—62no—
225#95#6712yes—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C3×Q8—41no—
229C3×Q8—41no—
230C3×D4—42no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×C3×C4C2×SD1611yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264Dic6—22no—
265Dic6—22no—
266Dic6—22no—
267Dic6—22no—
268Dic6—41no—
269Dic6—41no—
270Dic6—41no—
271Dic6—41no—
272C3×C8—81no—
273C3⋊C8—42no—
274C3⋊C8—42no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—32no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—122no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—124no—
303SD16—124no—
304SD16—124no—
305M16—121no—
306M16—121no—
307C2×C8—41no—
308C2×C8—62no—
309C2×C8—62no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—124no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C4⋊C4—121no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—121no—
326C42—121no—
327C42—121no—
328C42—121no—
329C42C2×D1211yescharacteristic
330C42#23511yescharacteristic
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3#39211yescharacteristic
339C22×C3#44811yescharacteristic
340C22×C3#44911yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4—81no—
347C3×C4#45011yescharacteristic
348C3×C4#45211yescharacteristic
349C3×C4#46511yescharacteristic
350C3×C4#46811yescharacteristic
351Dic3—42no—
352Dic3—42no—
353Dic3—81no—
354Dic3—81no—
355C23—41no—
356C23#93711yescharacteristic
357Q8—41no—
358Q8—41no—
359Q8—62no—
360Q8—62no—
361Q8—62no—
362Q8—62no—
363Q8—121no—
364Q8—121no—
365Q8—121no—
366Q8—121no—
367D4—42no—
368D4—41no—
369D4—41no—
370D4—81no—
371D4—81no—
372D4—81no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4#88211yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4#88411yescharacteristic
395C2×C4#91111yescharacteristic
396C2×C4#91211yescharacteristic
397C2×C4D3×SD1611yescharacteristic
398C2×C4#91511yescharacteristic
399C2×C4#93011yescharacteristic
400C2×C4#94011yescharacteristic
401C8—81no—
402C8—122no—
403C8—122no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—161no—
407C2×C3#200011yescharacteristic
408C2×C3#200911yescharacteristic
409C2×C3#201011yescharacteristic
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22#496011yescharacteristic
416C22#515811yescharacteristic
417C22#517011yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—81no—
424C4—122no—
425C4—122no—
426C4—241no—
427C4—241no—
428C4#477611yescharacteristic
429C4#477811yescharacteristic
430C4#478011yescharacteristic
431C4#478311yescharacteristic
432C3#999911yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—161no—
436C2#7005311yescharacteristic
437C2#7020511yescharacteristic
438C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1517D3
hypocenterC3#9999
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
3. #74191C2semidirect
5. #70770C2semidirect
6. #70766C2semidirect
7. #67631C2semidirect
143. C2×Dic6SD16semidirect
432. C3#9999semidirect

Maximal subgroups

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

subgroupindex
#833152
#833142
#741912
C3×#15172
#707702
#707662
#676312
#99993