Finite Group Databasebeta

← #10691172

Subgroups — #10691172

Order 768.

478 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 91 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#82977C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic, complemented
3C3×#1527C211yesmaximal, characteristic
4#71718C211yesmaximal, characteristic
5C4×#824C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9188—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5230C411yescharacteristic
14#5230C411yescharacteristic
15#5222C411yescharacteristic
16#5222C411yescharacteristic
17C2×#824—21no—
18#4949—21no—
19#4947—21no—
20#4944—21no—
21#4944—21no—
22#4937C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4828—22no—
26#4719—21no—
27#4718—21no—
28#4701—22no—
29C3×#329—22no—
30#4554—21no—
31#4540—22no—
32#2983—31no—
33#2632—31no—
34C4×#101—31no—
35#1527—31no—
36#1527D311yescharacteristic
37#1288—31no—
38#1288—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4C2×C411yescharacteristic
58C2×Dic3⋊C4C2×C411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95C2×C411yescharacteristic
76C3⋊#95C2×C411yescharacteristic
77#830—21no—
78#824—24no—
79#815—22no—
80Dic3⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C4×C3⋊C8—21no—
87C2×#124C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#446—62no—
90#446—62no—
91#432—31no—
92#432—61no—
93#432C2×D311yescharacteristic
94#431—61no—
95C2×#101—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#399—61no—
99#395—31no—
100#395—31no—
101#394—31no—
102#394—31no—
103#393—61no—
104#388—61no—
105#388—61no—
106#383—31no—
107#343—61no—
108#329—22no—
109#329—62no—
110C24×C3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—81no—
117C22×C3×C4—21no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C22×Dic3C22⋊C411yescharacteristic
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—22no—
130C2×Dic6—41no—
131C3×M16—22no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C3×C42#4111yescharacteristic
143#217—42no—
144#217—42no—
145#217—81no—
146#217—81no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—81no—
154Dic3⋊C4—81no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159#208—22no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×Q8—31no—
164C22×Q8C4×D311yescharacteristic
165C22×D4C4×D311yescharacteristic
166C2×M16—21no—
167C2×M16—61no—
168#124—21no—
169#124—21no—
170#124—21no—
171#124—21no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C42—31no—
188C2×C42—31no—
189C2×C42C22×D311yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—62no—
192C4⋊C8—62no—
193C4⋊C8—61no—
194#101—64no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—62no—
200#97—61no—
201#97#6712yes—
202C4×C8—21no—
203C4×C8—62no—
204C4×C8—61no—
205#95—31no—
206#95—31no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3C2×C22⋊C411yescharacteristic
219C3×Q8—81no—
220C3×Q8—81no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3#10412yes—
252Dic6—42no—
253Dic6—42no—
254C3×C8—42no—
255C3×C8—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C24—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—62no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—81no—
265C2×Q8—121no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279M16—22no—
280M16—41no—
281M16—62no—
282M16—121no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—121no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—241no—
294C4⋊C4—241no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C22⋊C4—241no—
302C22⋊C4—241no—
303C42—61no—
304C42—61no—
305C42—61no—
306C42—61no—
307C42#23511yescharacteristic
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#38611yescharacteristic
321C22×C3C2×#10411yescharacteristic
322C22×C3#42111yescharacteristic
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—81no—
330Dic3—22no—
331Dic3—41no—
332Dic3—41no—
333Dic3—41no—
334Dic3—81no—
335Dic3—81no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23D3×C22⋊C411yescharacteristic
344Q8—81no—
345Q8—81no—
346Q8—122no—
347Q8—122no—
348D4—81no—
349D4—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—32no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4#67⋊C411yescharacteristic
374C2×C4#88311yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4C4×#6711yescharacteristic
377C2×C4#93711yescharacteristic
378C2×C4#93811yescharacteristic
379C8—42no—
380C8—42no—
381C8—122no—
382C8—122no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—161no—
386C2×C3#187111yescharacteristic
387C2×C3#191211yescharacteristic
388C2×C3#191411yescharacteristic
389C22—21no—
390C22—21no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22D3×#10411yescharacteristic
399C22#482111yescharacteristic
400C22#495611yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—62no—
408C4—81no—
409C4—121no—
410C4—121no—
411C4—121no—
412C4—241no—
413C4—241no—
414C3#918811yescharacteristic, complemented, Sylow
415C2—21no—
416C2—21no—
417C2—161no—
418C2#7008911yescharacteristic
419C2#7038111yescharacteristic
420C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1527D3
hypocenterC3#9188
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #79314C2semidirect
5. C4×#824C2semidirect
6. #67752C2semidirect
7. #67666C2semidirect
414. C3#9188semidirect

Maximal subgroups

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

subgroupindex
#829772
#793142
C3×#15272
#717182
C4×#8242
#677522
#676662
#91883