Finite Group Databasebeta

← #10695981

Subgroups — #10695981

Order 768.

566 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 83 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#82383C211yesmaximal, characteristic, complemented
2#82310C211yesmaximal, characteristic
3#79318C211yesmaximal, characteristic, complemented
4C3×#1490C211yesmaximal, characteristic, complemented
5#69867C211yesmaximal, characteristic, complemented
6#69863C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#10005—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5222—21no—
15#5217—21no—
16#5212—21no—
17#5212—21no—
18C2×#825—22no—
19#5015—21no—
20#5014—22no—
21#5008—21no—
22#4985—21no—
23#4968—21no—
24#4945—22no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4926C2211yescharacteristic
27#4847—22no—
28#4736—21no—
29#4735—21no—
30#4713—21no—
31#4702—21no—
32C3×D4⋊C8—22no—
33#4465—22no—
34#3234—31no—
35#3109—31no—
36#2562—31no—
37#1980—31no—
38#1528—31no—
39#1490D311yescharacteristic, complemented
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×M16—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—22no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3D412yes—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#869D411yescharacteristic
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75C3×#97—21no—
76C3×C4×C8—21no—
77#832—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81#830—21no—
82#825—44no—
83#821—22no—
84#821—22no—
85#818—22no—
86#817—22no—
87#809—22no—
88#808—22no—
89C4⋊(C3⋊C8)—21no—
90#803D412yes—
91#532—31no—
92C2×#125—31no—
93C2×C4×D4C2×D311yescharacteristic
94#501—61no—
95#490—61no—
96#481—61no—
97#481—61no—
98#480—61no—
99#478—62no—
100#476—61no—
101#459—61no—
102#447—61no—
103#442—62no—
104#432—31no—
105#432C2×D311yescharacteristic
106C2×#104—62no—
107C4×M16C2×D311yescharacteristic
108#400—61no—
109#394—61no—
110#388—62no—
111#380—61no—
112#380—61no—
113C4×C4⋊C4—31no—
114#330—62no—
115D4⋊C8—22no—
116C24×C3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—41no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132#235—42no—
133#235—42no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C3×M16—22no—
140C3×M16—41no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—44no—
152#217—82no—
153#213—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—42no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—42no—
168C2×C3⋊C8—22no—
169C2×#41—61no—
170C22×D4D1211yescharacteristic
171C23×C4—21no—
172C2×D8—62no—
173C2×M16—21no—
174C4⋊Q8—31no—
175#127—31no—
176#125—61no—
177#125—61no—
178#125—61no—
179#125—61no—
180#124—61no—
181#121—61no—
182#121—61no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42—32no—
199C2×C42C22×D311yescharacteristic
200#107—62no—
201#106—62no—
202#106—62no—
203C4⋊C8—61no—
204C4⋊C8#6712yes—
205#104—124no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212#97—21no—
213#97—21no—
214#97—32no—
215C4×C8—21no—
216#95—62no—
217#95—62no—
218#95—61no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—41no—
231C3×D4—41no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—82no—
240C2×C3×C4—82no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247#67—81no—
248#67—81no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258D12—42no—
259D12—42no—
260D12—42no—
261C4×D3—82no—
262Dic6—42no—
263C3×C8—42no—
264C3×C8—42no—
265C3⋊C8—42no—
266C24—21no—
267#41—122no—
268#41—122no—
269C2×Q8—61no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—62no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—121no—
286C22×C4C2×D1211yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289D8—124no—
290M16—22no—
291M16—41no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—62no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—242no—
314C42—122no—
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39211yescharacteristic
329C22×C3#45311yescharacteristic
330C22×C3#45411yescharacteristic
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—81no—
340C3×C4#45412yes—
341Dic3—82no—
342Dic3—81no—
343Dic3—81no—
344Dic3—81no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—122no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—241no—
363D4—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—242no—
383C2×C4#88211yescharacteristic
384C2×C4#88411yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#93011yescharacteristic
387C2×C4#93711yescharacteristic
388C2×C4#94011yescharacteristic
389C8—42no—
390C8—42no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3#200311yescharacteristic
396C2×C3#200611yescharacteristic
397C2×C3#201011yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22#482211yescharacteristic
412C22#482711yescharacteristic
413C22#496011yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—242no—
421C4—241no—
422C4—241no—
423C4—241no—
424C4#520112yes—
425C3#1000511yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#6999311yescharacteristic
431C2#7032611yescharacteristic
432C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1490C2
socleC22×C3#392
prime radical#82310C2
hypercenter#1490D3
hypocenterC3#10005
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
1. #82383C2semidirect
3. #79318C2semidirect
4. C3×#1490C2semidirect
5. #69867C2semidirect
6. #69863C2semidirect
7. #67750C2semidirect
39. #1490D3semidirect
425. C3#10005semidirect

Maximal subgroups

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

subgroupindex
#823832
#823102
#793182
C3×#14902
#698672
#698632
#677502
#100053